An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations
An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional...
Raei, Marzieh;Cuomo, Salvatore
2021-10-19 00:00:00
axioms Article An Efficient Localized Meshless Method Based on the Space–Time Gaussian RBF for High-Dimensional Space Fractional Wave and Damped Equations 1 2, Marzieh Raei and Salvatore Cuomo * Department of Applied Mathematics, Malek Ashtar University of Technology, Tehran 158751774, Iran; marzie.raei@gmail.com Scuola Politecnica e delle Scienze di Base, University of Naples Federico II, 80138 Napoli, Italy * Correspondence: salvatore.cuomo@unina.it Abstract: In this paper, an efficient localized meshless method based on the space–time Gaussian radial basis functions is discussed. We aim to deal with the left Riemann–Liouville space fractional derivative wave and damped wave equation in high-dimensional space. These significant problems as anomalous models could arise in several research fields of science, engineering, and technology. Since an explicit solution to such equations often does not exist, the numerical approach to solve this problem is fascinating. We propose a novel scheme using the space–time radial basis function with advantages in time discretization. Moreover this approach produces the (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Therefore the local feature, as a remarkable and efficient property, leads to a sparse coefficient matrix, which could reduce the com- putational costs in high-dimensional problems. Some benchmark problems for wave models, both Citation: Raei, M.; Cuomo, S. An wave and damped, have been considered, highlighting the proposed method performances in terms Efficient Localized Meshless Method of accuracy, efficiency, and speed-up. The obtained experimental results show the computational Based on the Space–Time Gaussian capabilities and advantages of the presented algorithm. RBF for High-Dimensional Space Fractional Wave and Damped Keywords: space–time radial basis function; wave equation; damped wave equation; high-dimensional Equations. Axioms 2021, 10, 259. localized meshless method https://doi.org/10.3390/ axioms10040259 MSC: 65D15; 65D25; 65D18 Academic Editor: Filomena Di Tommaso Received: 31 August 2021 1. Introduction Accepted: 8 October 2021 In recent decades, scientists and researchers have paid much attention to the expres- Published: 19 October 2021 sion of the physical models and chemical processes in the form of fractional derivative equations. Over time, the development of fractional calculus theory has provided an Publisher’s Note: MDPI stays neutral advantageous tool for modelling many natural processes that often have complex and with regard to jurisdictional claims in anomalous modeling and cannot be easily expressed with classical derivative calculus. published maps and institutional affil- Thus, the popularity of fractional calculus theory increased because they can satisfacto- iations. rily model such problems, which especially have been occurred in physics, chemistry, quantum mechanics, and viscoelasticity [1,2]. From about 20 years, radial basis functions (RBFs) are a primary mesh-free method for numerically solving PDEs on with collocation approaches (see, for example [3–5]). The versatility of scattered data interpolation tech- Copyright: © 2021 by the authors. niques is confirmed by a lot of applications, e.g., surface reconstruction, image restoration Licensee MDPI, Basel, Switzerland. and inpainting, meshless/Lagrangian methods for fluid dynamics, surface deformation This article is an open access article or motion capture systems allowing the recording of sparse motions from deformable distributed under the terms and objects such as human faces and bodies [6]. The numerical solution of partial differential conditions of the Creative Commons equations by a global collocation approach based on RBF, is also referred to as a strong form Attribution (CC BY) license (https:// solution in the PDE literature [7–10]. An alternative interesting approach in collocation creativecommons.org/licenses/by/ methods is to use other bases as for example Hermite exponential spline defined in [11]. 4.0/). Axioms 2021, 10, 259. https://doi.org/10.3390/axioms10040259 https://www.mdpi.com/journal/axioms Axioms 2021, 10, 259 2 of 22 The main drawback in the global approach, though spectrally accurate, consists in solving large, ill-conditioned, dense linear systems and many attempts are known to deal with it [12,13]. Recently, a direct meshless method based on Gaussian radial basis functions has been developed to solve one-dimensional linear and nonlinear convection-diffusion problems [14]. Local methods are so preferred, giving up spectral accuracy for a sparse, better-conditioned linear system and more flexibility for handling non-linearities. A wide literature concerns local RBF schemes by partitioning the domain, referred to as Partition of Unity (PU) [15–17]. Moreover, the two-term time-fractional PDE model, one of the inter- esting models in mathematical physics, is investigated and solved by employing a local meshless method [18]. More recently, numerically, a local meshless collocation method has been applied to simulate the time-fractional coupled Korteweg-de Vries and Klein-Gordon equations [19]. This paper applies an efficient local meshless approach based on one of the most applicable and popular RBFs, the so-called Gaussian RBF, to solve the space fractional derivative equations. The proposed method, due to its local property, which creates the sparse coefficient matrix and accelerates the execution of the algorithm as well as uses the space–time RBFs, can be a suitable tool for solving practical problems in physics and engi- neering with space fraction derivatives. More in detail, one of the most popular fractional models is the space fractional wave equation created from the classical wave equation by replacing the integer space derivatives of order 2 with some fractional derivatives of order g, 1 < g 2. Finally, the numerical results demonstrate the power and efficiency of the suggested method to apply for more complex fractional partial differential equations with applications in high-dimensional computational finance. In this work, we consider a space fractional damped wave equation as follows: ¶ ¶ a a 1 2 u(x, t) + A u(x, t) + A u(x, t) = K D u(x, t) + K D u(x, t) + a a 1 2 1 x 2 x 1 1 2 2 ¶t ¶t +K D u(x, t) + f (x, t), n a n x with, x = (x , x , , x ) 2 W R , 0 < t T, (1) 1 2 n with the initial and boundary conditions ¶u u(x, 0) = f(x, y), (x, 0) = y(x, y), (x, y) 2 W, (2) ¶t u(x, t) = 0, (x, y) 2 ¶W, 0 < t T, (3) where u is the unknown function, f is the source function, W = W[ ¶W = [0, 1] , the co- efficients K , K , , K > 0. Moreover, 1 < a , a , , a 2 are the order of the 1 2 n 1 2 n a a a 1 2 n left Riemann–Liouville space fractional derivatives D u, D u, , D u respect to a a a x 1 x1 2 x2 n x , x , , x , respectively. The left and right Riemann–Liouville fractional derivative is 1 2 n defined by 1 d 1 a D u(x, t) = (x x) u(x , , x, , x , t)dx, (4) a x i 1 n i i G(2 a ) dx a and 2 b 1 d i 1 a D u(x, t) = (x x ) u(x , , x, , x , t)dx, (5) x n i 1 b 2 G(2 a ) dx i i where G denotes the gamma function. In the governing Equation (1), if A = A = 0, we have the space fractional wave 1 2 equation. In this work, both wave and damped wave equation are investigated. The paper is organized as follow. The Section 2 is devoted to the derivation of the numerical method for wave and damped equations. In the Section 3, numerical results in terms of accuracy and efficiency have been reported. Finally, Conclusions close the paper. Axioms 2021, 10, 259 3 of 22 2. The Numerical Scheme for Wave and Damped Models In this section, an impressive numerical method in the meshless literature with local features based on the space–time Gaussian radial basis function is investigated to solve the left Riemann–Liouville space fractional derivatives wave and damped wave equation in high-dimensional space. Due to using the space–time radial basis function, the suggested method does not need to discretize the problem in the time direction; this capability leads to (n + 1)-dimensional spatial-temporal computational domain for n-dimensional problems. Hence, by employing the local meshless technique as a significant feature [20], the resulting coefficient matrix will be sparse. This remarkable ability reduces computational costs and increases the speed-up of the numerical process for high-dimensional problems. Therefore the local feature, as a remarkable and efficient property, of the suggested numerical proce- dure leads to the sparse coefficient matrix, which could reduce the computational costs in high-dimensional problems. For this purpose, we need to evaluate and approximate the differential operators that appear in governing problem (1) based on the space–time Gaussian radial basis function. Then the localized meshless method is applied to solve the high-dimensional wave and damped wave equations. 2.1. Evaluate Derivative Operators with Space–Time Gaussian RBF Radial Basis Functions (RBFs) are a class of real-valued functions whose values at a specified data point depending on the distance from two points in multi-dimensional space. Kansa first used RBFs in 1991 to solve fluid dynamics problems [7]. Then, due to its ease of use, RBF was given special attention by scientists in science and engineering, and several meshless methods using RBFs were proposed. In recent decades, meshless numerical techniques attracted a lot of attention due to their high flexibility and good performance to deal with practical high dimensional models with complicated and irregular domains applied problems in science and engineering. Therefore, it is necessary to first provide the general definition of the function as follows n n Definition 1. Let R be n-dimensional Euclidean space. Let F : R ! R be an invariant function n n whose value at any point x 2 R depends only on the distance from the fixed point x 2 R , and can be written F(kx xk). (6) Then the function F is a radial basis function (RBF), where x is the center of the RBF F. Some popular and commonly used kinds of the RBFs include 2 2 c r • Gaussian: F(r) = e , 2 2 • Multiquadric: F(r) = r + c , • Inverse multiquadric: F(r) = ’ 2 2 r +c • Thin plate spline: r log r, where r = kx xk denotes the Euclidean distance between x and x. Moreover, c is the shape parameter which has an important rule to control the accuracy and stability of the numerical method. In the current work, the Gaussian RBF is used in the numerical experiments. Gaussian RBF is strictly positive definite function which causes the Gaussian RBF to be the most commonly used RBF. Aslo, Gaussian RBF is a representative member of the class of infinitely differentiable functions with global support. As mentioned earlier, we use space–time dependent radial basis functions. In the current work, the Gaussian RBF is used to approximate the numerical solution u in rela- tion (21), which is defined as 2 2 c k(x,t) (x ,t )k n i i F (x, t) = e , x 2 R , 0 < t T. (7) i Axioms 2021, 10, 259 4 of 22 Moreover, the Gaussian RBF could be described by the following form 2 2 c r F (r) = e , (8) where 2 2 2 2 r = (x x ) + (x x ) + + (x x ) + e(t t ) . (9) 1 1i 2 2i n ni i with (x , x , , x ) = x 2 R , 0 < t T. In the two above definitions of the Gaussian 1 2 function, the c and e constants are known as shape parameters. They play an important role in the accuracy and stability of computation. Finding the optimal value of shape parameters is a fundamental issue of meshless methods based on RBFs. We have used trial and error techniques to find the best value of shape parameters in the numerical procedure. As can be seen, Gaussian functions, due to their mathematical structure, have a special ability and ease of use to be applied to approximate fractional and integer derivative operators. Therefore the second integer derivative of u(x, t) respect to t could be approximated as follow 2 N 2 2 2 2 2 u(x, t) = 2c e (2c e (t t ) 1)F (x, t). (10) å i i ¶t i=1 The approximation of Riemann–Liouville spatial fractional derivatives using RBFs is a significant and major issue. Estimating and calculating such derivatives is not easy for most RBFs; however, selecting the Gaussian RBF allows us to use the Taylor expansion of this function to approximate Riemann–Liouville fractional derivatives in the x and y directions. The approximation of fractional derivatives due to the expansion of the gaussian in a MacLaurin series is investigated in [21]. In this work, we follow a similar way to approximate the space fractional derivative as follows 2 2 2 2 2 2 a a c [(x x ) ++(x x ) ++(x x ) +e (t t ) ] 1 1i j ji n ni i D F (x , , x , , x , t) = D e x i 1 j x j j 2 2 2 2 2 2 2 2 2 c [(x x ) , ,(x x ) +(x x ) ++(x x ) +e (t t ) ] c (x x ) 1 1i j 1 j 1i j+1 j+1i n ni i j ji = e D e , (11) where 2 p q 2 p ¥ p 2 p (2 p)!( 1) x 2 2 ( 1) c ji p c (x x ) a a j ji D e = D x (12) x å å x j j p! q!(2 p q)! p=0 q=0 Moreover, if the left Riemann–Liouville fractional derivative operator is considered, a a i.e., D := D then the left Riemann–Liouville fractional derivative of polynomials are x x j j defined by a q q k k q!(x a) G(a + 1) x ( a) q j D x = (13) x å G(q + 1 a) k!G(a k + 1) k=0 a a Moreover, for right Riemann–Liouville fractional derivative operator, i.e., D := D j b we have q k k q!(b x ) G(a + 1) j x ( b) D x = . (14) x å j b G(q + 1 a) k!G(a k + 1) k=0 2.2. Meshless Localized Space–Time RBF Collocation Method One of the most efficient meshless methods to deal with time-dependent problems is the localized space–time radial basis function (RBF) collocation method [22]. This technique as a generalization of radial basis function collocation method is defined by considering both the spatial and time variable to construct the radial basis functions. To better clarify, before introducing the localized RBF collocation method, the main idea of the global RBF Axioms 2021, 10, 259 5 of 22 collocation method is briefly reviewed. In general, we consider the following boundary value problem on global bounded domain W = W[ ¶W: Lu(x, t) = f(x, t), (x, t) 2 W, (15) Bu(x, t) = g(x, t), (x, t) 2 ¶W, (16) where L is the given differential operator in problem (1), which is defined as follows: ¶ ¶ a a 1 2 n L := + A + A K D K D K D , (17) 1 2 1 a x 2 a x n a 1 2 n x 2 1 2 n ¶t ¶t and B is the boundary operator in Equation (3). According to the global RBF collocation method, assume a set of scattered center points i N X = f(x , t )g in W and X = f(x , t )g on ¶W cover the entire global domain. I i i B i i i=1 i=N i+1 Then the unknown solution of the boundary value problem (15) and (16) is represented in the following form: u(x, t) ' l F(k(x, t) (x , t )k), (x, t) 2 W, (18) å i i i i=1 ¯ ¯ where F : W W ! R is a selected radial basis function,kk denotes the Euclidean norm and fl g are unknown coefficients to be determined. By Substituting the sugggested i=1 solution (18) in the Equation (15) with boundary condition (16), the collocation procedure conclude the algebraic system under the form: l LF(k(x , t ) (x , t )k) = f(x , t ), j = 1, , N , (19) å i j j i i j j i i=1 l BF(k(x , t ) (x , t )k) = g(x , t ), j = N , , N, (20) å i j j i i j j i+1 i=1 By solving the system of Equations (19) and (20) the unknown vector fl g is de- i=1 termined. Then the solution u at any point of the entire computational domain can be determined by substituting the vector fl g in the Equation (18). The meshless global i=1 RBF collocation method, also known in the literature as the Kansa method, was introduced by Kansa in two famous references [7] and got some attention from researchers. The global collocation method, in turn, has been proposed as an efficient technique in dealing with high-dimensional problems and solving problems in complex computational domains. Despite all the advantages of the global collocation method, the coefficient matrix is dense and imposing high computational costs. Moreover, the coefficients matrix is generally ill-conditioned and leading to ill-conditioning behaviour in the numerical method. In addi- tion, the numerical solution is seriously affected by the shape parameter in the radial basis functions dependent on the shape parameter such as Gaussian and multiquadric RBFs, so selecting the appropriate shape parameter is difficult and sensitive. Due to the popularity of meshless methods, various techniques have been proposed to overcome these difficulties and reduce their destructive effects. One of the most popular numerical techniques is the local collocation method developed by Lee et al. They demonstrated that the local colloca- tion method is less sensitive to the selection of the shape parameter and the distribution of scattered points in the computational domain, but the computational accuracy is slightly reduced compared to the global method. Moreover, the resulting final coefficients matrix in the local collocation method is sparse and considerably well-conditioned. In the local collocation method, for any center point (x , t ) in the computational domain W, a local c c sub-domain W consist of n nearest neighbor points is considered. The set of points in c c the local sub-domain is called a stencil. The approximate solution u(x , t ) using the RBF c c Axioms 2021, 10, 259 6 of 22 collocation method on the stencil W can be obtained by a linear combination of the radial basis functions at n nearest neighbouring points of (x , t ) in the following form: c c c u(x , t ) ' u(x , t ) = l F(k(x , t ) (x , t )k), (x , t ) 2 W , (21) c c c c c c c å k k k k k k=1 where F is a radial basis function and fl g are unknown coefficients to be determined. k=1 Since f(x , t )g W from Equation (21) the following linear system is obtained k k k=1 U = F L , (22) c c c where F = [F(k(x , t ) (x , t )k)], 1 i, j n , c c i i j j U = [u ˆ(x , t ), u ˆ(x , t ), , u ˆ(x , t )], c n n 1 1 2 2 c c L = [l , l , , l ] . c 1 2 n Therefore, solving the linear system (22) yields the unknown coefficients as follows: L = F U . (23) c c Evaluating the operator L of the interpolant (21) and using the relation (23) give Lu ˆ(x , t ) = l LF(k(x , t ) (x , t )k) (24) c c c c å k k k k=1 = [LF(k(x , t ) (x , t )k), ,LF(k(x , t ) (x , t )k)]L (25) c c 1 1 c c k k = (LF )F U . (26) c c Now, by substituting the derivative operatorL in relation (17), the following equation is obtained ¶ ¶ a a a 1 2 [ + A + A K D K D K D ]u(x , t ) 1 2 1 a x 2 a x n a c c n x 1 1 2 2 n ¶t ¶t ¶ ¶ a a a 1 1 2 n = ([ + A + A K D K D K D ]F )F U 1 2 1 a x 2 a x n a c c 1 2 2 n x c 2 1 n ¶t ¶t | {z } cL = W U . (27) cL As the same way, for all (x , t ) 2 W the weight vector W = [W , , W ] is j j L 1L N L computed such that Lu ˆ(x, t) = W U , (x, t) 2 W, (28) L L ˆ ˆ where U = [u(x , t ), , u(x , t )]. By substituting the relation (28) in Equation (15), L 1 1 N N i i we have W U = f(x, t), (x, t) 2 W. (29) L L We also consider a local computational domain for each point on the boundary of computational domain similar to that discussed at points within the computational domain. Therefore, for each point (x , t ) 2 ¶W, according to the collocation method, the effect of c c the operator B on the function u is approximated as follows: Bu ˆ(x , t ) = l BF(k(x , t ) (x , t )k) (30) c c å k c c k k k=1 = [BF(k(x , t ) (x , t )k)BF(k(x , t ) (x , t )k)]L (31) c c c c 1 1 k k = (BF )F U . (32) c c c Axioms 2021, 10, 259 7 of 22 Given that the boundary condition’ operator is the identity operator, the following equation has resulted u(x , t ) = F F U c c c c | {z } cB = W U . (33) cB c Moreover, for all (x , t ) 2 ¶W the weight vector W = [W , , W ] is calcu- j j B N B NL i+1 lated as follows: u ˆ(x, t) = W U , (x, t) 2 ¶W, (34) B B where U = [u ˆ(x , t ), , u ˆ(x , t )], By substituting the Equation (34) in Equa- B N N N N i+1 i+1 tion (16), the following relation is obtained W U = g(x, t), (x, t) 2 ¶W. (35) B B Then, the following linear system of equations with N N global sparse matrix W = [W ; W ] for all center points f(x , t )g , could be assembled such that L B j j j=1 f(x , t ) W U L L i i = , 1 i N , N j N. (36) i i+1 W U g(x , t ) B B j j Then the approximate solution fu(x , t )g can be obtained by solving the above i i i=1 sparse linear system of equations. 3. Numerical Results In this section, some test problems in one, two, and three spatial dimensional for investigating the accuracy and efficiency of the presented method for both wave and damped wave equations are considered. As mentioned in the suggested technique the space–time RBFs are considered, therefore in the computational process for n-dimensional spatial model, we consider the (n + 1)-dimensional spatial-temporal computational domain with uniform distributed points. The scheme of computational domain W = [0, 1] with scattered data points for one and two-dimensional are demonstrated in Figure 1. The three- dimensional model leads to the four-dimensional computational domain and it could not be shown. Moreover, two numerical criteria to show the accuracy, convergence, and stability of the method as absolute error (e ) and the root mean square error (e ) are considered as ¥ r follows: ku(x, t) u ˆ(x, t)k e = ku(x, t) u ˆ(x, t)k , e = p . (37) ¥ ¥ r (a) (b) Figure 1. Data location scheme for (a) Example 1 and (b) Example 2. Axioms 2021, 10, 259 8 of 22 Moreover, to investigate the convergence rates of the presented discretization scheme the following rate is estimated: log (e (h )/e (h )) ¥ ¥ 2 R = , log (h /h ) 1 2 where h = 1/N. Moreover, in numerical implementation for constructing the local sub- domains, the k-dimensional tree (k-d tree) algorithm is used [23]. To improve search performance, the search tree structure is first created. Then for each computational point in the spatial-temporal domain, the k-d tree algorithm search n nearest neighbors. More- over, to investigate the numerical stability of the suggested method to deal with the mentioned models, the noise efficacy on computational error estimates is perused. To this end, we assume that the initial solution u in the numerical procedure is perturbed to u ˆ = (1 + s)u . Thus, the impression of the input noise s on error estimates is studied for 0 0 perturbation solution u ˆ. Example 1. As the first example, we consider the one-dimensional space fractional derivatives problem (1) with corresponding to wave model by coefficients K = 10, A = A = 0 and damped 1 1 2 wave model by coefficients K = 10, A = A = 0.2. The exact solution of the problem for both 1 1 2 models is u(x, t) = sin(p(t + 1))(x x ). The source functions for both models are calculated as follows 2 2 f (x, t) = p sin(p(t + 1)) + A p cos(p(t + 1)) + A sin(p(t + 1)) (x x ) 1 a 2 a 1 1 x 2x K sin(p(t + 1)) . G(2 a ) G(3 a ) 1 1 In one-dimensional problem the computational domain is W T where W = [0, 1] with uniform distribution data points and T = [0, 1]. Moreover, the local sub-domains are determined with k-d tree algorithm with size n = 5. Table 1 is demonstrated the computational errors, condition numbers, computational process times, and convergence rates concerning the different numbers of data points N and several values of fractional orders a for both wave and damped wave models. The numerical results verify that the presented method is accurate, convergent and, well-posed due to the increasing number of data points in the computational domain. The best value of the shape parameter c is determined by trial and error so that there is a balance between the obtained accuracy and the conditional number of the coefficient matrix. Therefore, the plot of absolute errors and condition numbers concerning the shape parameter c by letting N = 20 and a = 1.80 is demonstrated in Figure 2 for both wave and damped wave models. Moreover, the estimated errors for different values of N and various fractional orders a by taking N = 30 and shape parameter c = 0.015 for wave model and c = 0.010 for 1 x damped wave model are reported in the Table 2. The demonstrated results in this table show the temporal convergence of the presented procedure. The stability of the suggested method is verified numerically in Table 3 by considering the different input noise values s = 0, s = 0.001, s = 0.01, and s = 0.1. This table shows the influence of noise s on computational errors and presents that the proposed method has a reasonable and stable behavior against input noise. The Figures 3 and 4 are shown the absolute errors for different time levels t = 0.25, t = 0.50, and t = 0.75 by letting N = 41, shape parameter c = 0.007, and a = 1.65 for both wave and damped wave model, x 1 respectively. Moreover, Figures 5 and 6 are the plots of the exact and approximated solutions for wave and damped wave problems in the spatial-temporal computational domain, respectively. The sparsity patterns of the coefficient matrix related to both wave and damped wave models are plotted in Figure 7. To investigate the method’s stability numerically, the impression of the input noise on the initial solution is evaluated. For this purpose, effect of the several input noise levels s = 0, s = 0.001, s = 0.01, and s = 0.1 on error estimates are considered. These results are reported in Table 4 and illustrated the acceptable and stable behavior of the numerical procedure against the input noise. Axioms 2021, 10, 259 9 of 22 Table 1. Error estimates, condition numbers, CPU times and convergence rate of Example 1 by letting n = 5 and different values of N and several fractional orders a . Wave Model Damped Wave Model a N c e e CN Time R c e e CN Time R ¥ r ¥ ¥ r ¥ 2 3 4 3 3 4 3 1.80 5 0.006 2.2269 10 9.0263 10 5.1233 10 0.11 0.006 2.2934 10 9.0038 10 5.0633 10 0.14 2 4 4 5 4 4 5 10 0.007 4.0344 10 1.9577 10 1.2380 10 0.16 2.46 0.007 4.0829 10 1.9889 10 1.2371 10 0.18 2.48 2 4 5 6 4 5 6 15 0.008 1.5062 10 7.7824 10 1.5415 10 0.24 2.42 0.008 1.5571 10 7.9075 10 1.4739 10 0.28 2.37 2 5 5 6 5 5 6 20 0.009 6.9551 10 3.7304 10 2.8412 10 0.38 2.68 0.009 7.3215 10 3.8292 10 3.2369 10 0.40 2.62 2 5 5 7 5 5 6 25 0.010 3.6964 10 1.8750 10 1.8328 10 0.53 2.83 0.010 3.8836 10 1.9777 10 1.5684 10 0.58 2.84 2 5 5 7 5 5 7 30 0.015 2.4838 10 1.1746 10 1.0483 10 0.73 2.18 0.012 2.2484 10 1.1545 10 1.1188 10 0.80 2.99 2 3 3 3 3 3 3 1.60 5 0.006 3.3795 10 1.4492 10 4.5845 10 0.11 0.006 3.1493 10 1.4469 10 3.9339 10 0.12 2 4 4 5 4 4 5 10 0.007 6.5764 10 3.1243 10 1.5830 10 0.16 2.36 0.007 7.0697 10 3.1627 10 1.5540 10 0.17 2.15 2 4 4 6 4 4 6 15 0.008 2.4993 10 1.2304 10 1.5025 10 0.25 2.38 0.008 2.5861 10 1.2397 10 1.4145 10 0.26 2.48 2 4 5 7 4 5 6 20 0.009 1.1825 10 5.8391 10 2.1201 10 0.37 2.60 0.009 1.1111 10 5.7728 10 1.8260 10 0.41 2.93 2 5 5 7 5 5 6 25 0.010 6.0763 10 2.9338 10 7.0838 10 0.54 2.98 0.010 4.9088 10 2.7473 10 6.9984 10 0.55 3.66 2 5 5 7 5 5 7 30 0.015 5.6002 10 2.1354 10 9.0419 10 0.73 2.18 0.012 2.9940 10 1.4093 10 5.8078 10 0.77 2.99 2 3 4 3 3 4 3 1.90 5 0.006 1.8605 10 7.5008 10 3.6377 10 0.11 0.005 1.8677 10 7.4356 10 3.6238 10 0.12 2 4 4 6 4 4 5 10 0.007 4.7886 10 1.7566 10 1.1151 10 0.16 1.95 0.006 7.0447 10 2.4132 10 9.7984 10 0.17 1.40 2 4 5 6 4 4 6 15 0.008 1.5025 10 7.4091 10 1.9605 10 0.24 2.85 0.007 2.7777 10 1.1673 10 3.1303 10 0.27 2.29 2 5 5 6 4 5 6 20 0.009 7.8533 10 3.8566 10 8.9269 10 0.36 2.25 0.008 1.2280 10 5.2598 10 9.5888 10 0.38 2.67 2 5 5 7 5 5 7 25 0.010 5.0342 10 2.3522 10 1.4118 10 0.53 1.99 0.009 7.6035 10 3.2475 10 1.0856 10 0.55 2.14 2 5 5 7 5 5 7 30 0.015 4.1866 10 1.9279 10 2.7048 10 0.74 1.01 0.010 5.2990 10 2.2894 10 3.1228 10 0.76 1.98 Axioms 2021, 10, 259 10 of 22 (a) (b) Figure 2. Absolute errors and condition numbers for Example 1 by letting N = 20 and a = 1.80 for different shape parameter c with respect to (a) wave model and (b) damped wave model. Table 2. Error estimates for N = 30 of Example 1 by letting c = 0.015 in wave model and c = 0.010 in damped wave model, n = 5, different values of N and several fractional orders a . c t Wave Model Damped Wave Model a N 1 t e e e e ¥ r ¥ r 2 2 2 2 1.55 15 3.8035 10 1.3685 10 5.2703 10 2.5408 10 4 5 4 5 20 1.4421 10 6.3442 10 1.2401 10 6.3900 10 5 5 5 5 25 8.4107 10 3.8606 10 5.5679 10 3.2129 10 5 5 5 5 30 6.3251 10 3.0383 10 3.3992 10 1.7063 10 2 2 2 2 1.65 15 2.7988 10 1.2182 10 4.4133 10 2.1276 10 5 5 5 5 20 8.1961 10 3.9625 10 8.5992 10 4.7289 10 5 5 5 5 25 4.6205 10 2.0393 10 4.1815 10 2.3432 10 5 5 5 5 30 4.3104 10 1.7328 10 2.6376 10 1.2837 10 2 2 2 2 1.75 15 3.4041 10 1.5951 10 3.7402 10 1.8022 10 5 5 5 5 20 8.0199 10 3.3220 10 8.2582 10 3.9420 10 5 5 5 5 25 3.5874 10 1.6133 10 3.7671 10 1.9878 10 5 5 5 5 30 3.2914 10 1.3236 10 2.2795 10 1.1375 10 Table 3. The effect of noise on error estimates of Example 1 for different values of N and letting a = 1.75. Wave Model Damped Wave Model N s c e e c e e ¥ r ¥ r 2 4 4 4 4 10 0 0.007 4.5573 10 2.1751 10 0.008 4.5430 10 2.1807 10 4 4 4 4 0.001 6.9889 10 3.3333 10 6.6901 10 3.3199 10 3 3 3 3 0.01 2.8873 10 1.3784 10 2.8574 10 1.3744 10 2 2 2 2 0.1 2.4771 10 1.1834 10 2.4742 10 1.1829 10 2 4 5 4 5 15 0 0.008 1.6829 10 8.5675 10 0.009 1.6508 10 8.5114 10 4 4 4 4 0.001 4.1572 10 2.0592 10 3.9821 10 2.0389 10 3 3 3 3 0.01 2.6657 10 1.2901 10 2.6428 10 1.2873 10 2 2 2 2 0.1 2.5165 10 1.2134 10 2.5142 10 1.2131 10 2 5 5 5 5 20 0 0.009 8.2212 10 4.0938 10 0.010 8.4842 10 4.0896 10 4 4 4 4 0.001 3.2011 10 1.6265 10 3.0582 10 1.6079 10 3 3 3 3 0.01 2.5562 10 1.2661 10 2.5419 10 1.2638 10 2 2 2 2 0.1 2.4917 10 1.2304 10 2.4902 10 1.2301 10 2 5 5 5 5 25 0 0.010 3.9177 10 1.9537 10 0.011 3.7677 10 1.9476 10 4 4 4 4 0.001 2.7922 10 1.4207 10 2.7153 10 1.4024 10 3 3 3 3 0.01 2.5277 10 1.2572 10 2.5185 10 1.2552 10 2 2 2 2 0.1 2.5027 10 1.2411 10 2.5018 10 1.2409 10 Axioms 2021, 10, 259 11 of 22 (a) (b) (c) Figure 3. Absolute errors for Example 1 by letting N = 41, c = 0.007, and a = 1.65 for different time level t (a) t = 0.25; (b) t = 0.50 and (c) t = 0.75 with respect to wave model. (a) (b) (c) Figure 4. Absolute errors for Example 1 by letting N = 41, c = 0.007, and a = 1.65 for different x 1 time level t (a) t = 0.25; (b) t = 0.50 and (c) t = 0.75 with respect to damped wave model. Axioms 2021, 10, 259 12 of 22 (a) (b) Figure 5. The plots of (a) exact solution and (b) approximation solution for Example 1 by taking N = 41, c = 0.007, and a = 1.65 with respect to wave model. x 1 (a) (b) Figure 6. The plots of (a) exact solution and (b) approximation solution for Example 1 by taking N = 41, c = 0.007, and a = 1.65 with respect to damped wave model. x 1 (a) (b) Figure 7. The sparsity pattern of the coefficient matrix for Example 1 with respect to (a) wave model (b) damped wave model. Axioms 2021, 10, 259 13 of 22 Table 4. The effect of noise on error estimates of Example 4 for different values of N and letting a = 1.85, a = 1.65 and a = 1.45. 1 2 3 Wave Model Damped Wave Model N s c e e c e e ¥ r ¥ r 4 5 5 5 5 5 0 0.030 6.9620 10 1.2229 10 0.010 5.9004 10 1.0412 10 5 5 5 5 0.001 8.5245 10 1.4963 10 7.4629 10 1.3117 10 4 5 4 5 0.01 2.2587 10 3.9592 10 2.1525 10 3.7679 10 3 4 3 4 0.1 1.6321 10 2.8600 10 1.6215 10 2.8406 10 4 5 6 5 6 7 0 0.035 2.7000 10 5.7760 10 0.014 2.4669 10 5.2211 10 5 6 5 6 0.001 4.2625 10 8.9208 10 4.0294 10 8.3482 10 4 5 4 5 0.01 1.8325 10 3.7324 10 1.8091 10 3.6730 10 3 4 3 4 0.1 1.5895 10 3.2153 10 1.5871 10 3.2093 10 4 5 6 5 6 9 0 0.040 1.6106 10 3.3205 10 0.018 1.7590 10 3.1765 10 5 6 5 6 0.001 3.1488 10 6.4778 10 3.2283 10 6.4089 10 4 5 4 5 0.01 1.7211 10 3.6879 10 1.7290 10 3.6873 10 3 4 3 4 0.1 1.5783 10 3.4274 10 1.5791 10 3.4275 10 4 6 6 6 6 11 0 0.045 8.8645 10 1.9753 10 0.020 7.6517 10 1.7709 10 5 6 5 6 0.001 2.1782 10 5.0290 10 2.3160 10 5.2187 10 4 5 4 5 0.01 1.6139 10 3.6821 10 1.6378 10 3.7177 10 3 4 3 4 0.1 1.5676 10 3.5682 10 1.5700 10 3.5720 10 Example 2. As the second example (see Figure 8), the two-dimensional space fractional derivatives problem (1) related to wave model by coefficients K = K = 10, A = A = 0 and damped wave 1 2 1 2 model by by coefficients K = K = 10, A = A = 0.2. The exact solution of the problem for 1 2 1 2 2 2 either wave and damped wave model is u(x, y, t) = sin(p(t + 1))(x x )(y y ). The source functions for both models are obtained as follows f (x, y, t) = p sin(p(t + 1)) + A p cos(p(t + 1)) + A sin(p(t + 1)) 1 2 2 2 (x x )(y y ) 1 a 2 a 1 1 x 2x K sin(p(t + 1))(y y ) G(2 a ) G(3 a ) 1 1 1 a 2 a 2 2 y 2y K sin(p(t + 1))(x x ) . G(2 a ) G(3 a ) 2 2 In two-dimensional case the computational domain is W T where W = [0, 1] with uniform distributed points and T = [0, 1]. Moreover, the k-d tree program constructs the local sub- domains with size n = 10. The accuracy, convergence, and wellposedness of the presented procedure for solving both wave and damped wave models have shown in Table 5 by computing the estimated errors, conditional numbers, CPU times, and convergence rates for several values of N and various fractional orders a and a . The optimal value of the shape parameter c is specified by trial and error such that 1 2 there is an equivalence between the accuracy of the numerical method and the conditional number of the coefficient matrix. Thus, the plot of absolute errors and condition numbers with respect to the shape parameter c by taking N = 10 and a = a = 1.50 is shown in Figure 9 for both 1 2 wave and damped wave models. Moreover, the computational errors for various N and different fractional orders a and a by letting N = 20 and shape parameter c = 0.017 for wave model 1 1 and c = 0.020 for damped wave model are shown in the Table 6. The reported numerical results in this table demonstrate the temporal convergence of the proposed method in deal with both mentioned models. The behavior of the numerical method for either wave and damped wave model against the input noise is demonstrated in Table 7. The computed results show the stability of the presented method to deal with different noise levels. The absolute errors for several time levels by considering N = 21 , a = 1.65, a = 1.85 for wave model with shape parameter c = 0.022 and damped x 1 2 wave model with shape parameter c = 0.020 are plotted in Figures 10 and 11, respectively. The Axioms 2021, 10, 259 14 of 22 sparsity patterns of the coefficient matrix for both wave and damped wave models are demonstrated in Figure 7. Table 5. Error estimates, condition numbers, CPU times convergence rate of Example 2 by letting n = 10 and different values of N and several fractional orders a and a . 1 2 Wave Model Damped Wave Model (a , a ) N 1 2 c e e CN Time R c e e CN Time R ¥ r ¥ ¥ r ¥ 3 4 4 3 4 4 3 (1.5, 1.5) 5 0.006 4.8054 10 1.3983 10 2.4132 10 0.14 0.008 4.8850 10 1.3968 10 2.9502 10 0.17 3 4 5 6 4 5 6 10 0.009 1.5822 10 3.8986 10 1.1431 10 1.08 1.60 0.009 2.4934 10 6.3957 10 1.0003 10 1.24 0.97 3 5 5 6 4 5 7 15 0.011 5.6943 10 1.7840 10 9.0793 10 3.83 2.52 0.011 1.3405 10 4.1633 10 1.0522 10 3.93 1.53 3 5 5 8 5 5 8 20 0.016 3.9691 10 1.1626 10 2.4709 10 9.95 1.25 0.016 6.7846 10 2.0873 10 3.2388 10 10.36 2.36 3 4 4 3 4 4 3 (1.4, 1.8) 5 0.008 3.7221 10 1.0343 10 2.4408 10 0.17 0.009 3.7560 10 1.0388 10 2.8029 10 0.19 3 5 5 6 5 5 6 10 0.010 7.1562 10 2.2577 10 1.0923 10 1.02 2.37 0.010 7.8547 10 2.2794 10 1.6189 10 1.03 2.25 3 5 6 7 5 6 7 15 0.013 3.1481 10 8.5817 10 2.3637 10 3.80 2.18 0.013 3.2378 10 8.8020 10 2.6136 10 3.86 2.23 3 5 6 8 5 6 8 20 0.020 2.4336 10 4.4087 10 3.3440 10 10.03 1.41 0.020 2.1563 10 4.4523 10 4.5045 10 10.20 2.08 3 4 5 3 4 5 3 (1.9, 1.5) 5 0.008 3.1363 10 8.4538 10 2.5236 10 0.17 0.009 3.1664 10 8.5260 10 2.8063 10 0.18 3 5 5 5 5 5 5 10 0.010 6.1541 10 1.8819 10 9.4970 10 1.03 2.34 0.010 6.6609 10 1.8938 10 9.1129 10 1.04 2.24 3 5 6 6 5 6 6 15 0.013 2.3758 10 7.1676 10 8.5077 10 3.58 2.34 0.013 2.4072 10 7.2265 10 8.3153 10 3.84 2.51 3 5 6 7 5 6 7 20 0.020 1.2159 10 3.3100 10 3.9961 10 10.08 2.32 0.020 1.2487 10 3.3356 10 3.7771 10 10.21 2.28 (a) (b) Figure 8. The sparsity pattern of the coefficient matrix for Example 2 with respect to (a) wave model (b) damped wave model. (a) (b) Figure 9. Absolute errors and condition numbers for Example 2 by letting N = 10 and a = a = 1.50 for different shape parameter c with respect to (a) wave model and (b) damped 1 2 wave model. Axioms 2021, 10, 259 15 of 22 Table 6. Error estimates for N = 20 of Example 2 by letting c = 0.017 for wave model and c = 0.020 for damped wave model, n = 10, different values of N and several fractional orders a and a . c t 1 2 Wave Model Damped Wave Model (a , a ) N 1 2 t e e e e ¥ r ¥ r 3 3 2 3 (1.80, 1.70) 5 4.7343 10 1.4830 10 8.3832 10 1.6320 10 3 3 3 3 10 4.7001 10 1.5718 10 4.7465 10 1.5801 10 5 6 5 6 15 1.6565 10 4.9223 10 1.6170 10 4.7275 10 6 6 6 6 20 9.2334 10 2.8096 10 9.9772 10 3.3610 10 3 3 3 3 (1.60, 1.90) 5 4.5883 10 1.4206 10 5.7820 10 1.4226 10 3 3 3 3 10 4.5549 10 1.4870 10 4.6549 10 1.5537 10 5 6 5 6 15 1.7864 10 5.5979 10 1.6792 10 5.2939 10 5 6 5 6 20 1.0684 10 2.9493 10 1.0694 10 3.2222 10 3 3 3 3 (1.90, 1.90) 5 3.5657 10 1.0914 10 3.7033 10 1.2397 10 3 3 3 3 10 3.5159 10 1.1562 10 3.6942 10 1.2102 10 5 6 5 6 15 1.7359 10 4.9861 10 1.6265 10 4.7185 10 5 6 6 6 20 1.0300 10 2.7453 10 9.8631 10 2.5639 10 Table 7. The effect of noise on error estimates of Example 2 for different values of N and letting a = 1.85 and a = 1.45. Wave Model Damped Wave Model N s c e e c e e ¥ r ¥ r 3 4 5 4 5 5 0 0.008 3.4082 10 9.3084 10 0.009 3.4392 10 9.3683 10 4 4 4 4 0.001 4.0332 10 1.0973 10 4.0642 10 1.1020 10 4 4 4 4 0.01 9.6582 10 2.6043 10 9.6892 10 2.6049 10 3 3 3 3 0.1 6.5908 10 1.7720 10 6.5939 10 1.7718 10 3 5 5 5 5 10 0 0.010 6.4696 10 2.0480 10 0.010 6.9680 10 2.1079 10 5 5 4 5 0.001 1.2473 10 4.0433 10 1.2972 10 4.0640 10 4 4 4 4 0.01 6.6509 10 2.2138 10 6.7008 10 2.2125 10 3 3 3 3 0.1 6.0687 10 2.0322 10 6.0736 10 2.0321 10 3 5 6 5 5 15 0 0.015 3.1420 10 7.6784 10 0.015 6.8234 10 1.0182 10 5 5 5 5 0.001 8.2470 10 2.8254 10 8.4454 10 2.8972 10 4 4 4 4 0.01 6.4478 10 2.1932 10 6.4691 10 2.1934 10 3 3 3 3 0.1 6.2697 10 2.1320 10 6.2719 10 2.1319 10 3 5 6 5 6 20 0 0.020 1.2032 10 3.5863 10 0.020 1.2386 10 3.6963 10 5 5 5 5 0.001 6.7756 10 2.3326 10 6.8420 10 2.3303 10 4 4 4 4 0.01 6.2523 10 2.1953 10 6.2589 10 2.1948 10 3 3 3 3 0.1 6.2000 10 2.1837 10 6.2006 10 2.1836 10 Axioms 2021, 10, 259 16 of 22 (a) (b) (c) Figure 10. Absolute errors for Example 2 by letting N = 21 , c = 0.107, a = 1.65, and a = 1.85 for x 2 different time level t (a) t = 0.25; (b) t = 0.50 and (c) t = 0.75 with respect to wave model. (a) (b) (c) Figure 11. Absolute errors for Example 2 by letting N = 21 , c = 0.02, a = 1.65, and a = 1.85 1 2 different time level t (a) t = 0.25; (b) t = 0.50 and (c) t = 0.75 with respect to damped wave model. Axioms 2021, 10, 259 17 of 22 Example 3. As the third example, the two-dimensional space fractional derivatives problem (1) related to wave model by coefficients K = K = 10 , A = A = 0 and damped wave model by 1 2 1 2 coefficients K = K = 10 , A = 10 and A = 20. The exact solution of the problem for either 1 2 1 2 6 3 3 2 wave and damped wave model is u(x, y, t) = t (x x )(y 2y + y). The source functions for both models are obtained as follows 4 5 6 3 3 2 f (x, y, t) = 30t + 6 A t + A t (x x )(y 2y + y) 1 2 a 2 a 1 1 x 6x 6 3 2 K t (y 2y + y) G(1 a ) G(3 a ) 1 1 2 a 1 a a 2 2 2 6y 4y y 6 3 K t (x x ) + . G(3 a ) G(2 a ) G(1 a ) 2 2 2 In this case the computational domain is W T where W = [0, 1] with uniform and irregular distributed points and T = [0, 1]. Moreover,the k-d tree program constructs the local sub- domains with size n = 10. In the current example, the accuracy and efficiency of the proposed method in the two modes of uniform point distribution and irregular point distribution in the computational domain are compared for both wave and damped wave models. The irregular distribution points scheme is demonstrated in Figure 12. Figure 12. Data location scheme for Example 3 with irregular distribution. Figure 13 shows the plot of absolute estimated error concerning the number of uniform and irregular distribution data points for both uniform and irregular distributed data points.As can be seen, the accuracy of the suggested method in confronting the irregular distribution points is relatively good, but in comparison with the uniform distribution points, it Significantly decreases. Moreover, the condition numbers versus the number of uniform and irregular distributed data points for both wave and damped wave models have demonstrated in Figure 14. These Figures show that the present numerical method for irregular distribution points is not as good as uniform distribution points according to the condition number of the coefficient matrix. The sparsity pattern of the coefficient matrix for wave and damped wave models respect to the uniform and irregular distribution points are displayed in Figures 15 and 16, respectively. These figures demonstrate that the arising coefficient matrix is sparsity for both types of point distributions. Therefore, the presented method causes to decrease in the computational cost and accelerates the algorithm for irregular distribution points as good as uniform distribution points. Axioms 2021, 10, 259 18 of 22 (a) (b) Figure 13. The absolute errors vs. the number of data points for Example 3 with respect to (a) wave model (b) damped wave model. (a) (b) Figure 14. The condition numbers vs. the number of data points for Example 3 with respect to (a) wave model (b) damped wave model. (a) (b) Figure 15. The sparsity pattern of the coefficient matrix for Example 3 with respect to wave model with (a) uniform distribution of points (b) irregular distribution of points. Axioms 2021, 10, 259 19 of 22 (a) (b) Figure 16. The sparsity pattern of the coefficient matrix for Example 3 with respect to damped wave model with (a) uniform distribution of points (b) irregular distribution of points. Example 4. As the last example, the three-dimensional space fractional derivatives problem (1) for wave model by coefficients K = K = K = 10, A = A = 0 and damped wave model 1 2 3 1 2 by coefficients K = K = K = 10, A = A = 0.2. The exact solution of the problem is 1 2 3 1 2 2 2 2 u(x, y, t) = sin(p(t + 1))(x x )(y y )(z z ) The source functions for both models are concluded as follows f (x, y, z, t) = p sin(p(t + 1)) + A p cos(p(t + 1)) + A sin(p(t + 1)) 1 2 2 2 2 (x x )(y y )(z z ) 1 a 2 a 1 1 x 2x 2 2 K sin(p(t + 1))(y y )(z z ) G(2 a ) G(3 a ) 1 1 1 a 2 a 2 2 y 2y 2 2 K sin(p(t + 1))(x x )(z z ) G(2 a ) G(3 a ) 2 2 1 a 2 a 3 3 z 2z 2 2 K sin(p(t + 1))(x x )(y y ) . G(2 a ) G(3 a ) 3 3 In three-dimensional case the computational domain is W T where W = [0, 1] with uniform distributed points and T = [0, 1]. Further, in the computational procedure the local sub-domains with size nc = 15 are built by k-d tree algorithm. The error estimates, condition numbers, computational times, and convergence rates vs. N are reported in Table 8. The numerical results in this table are obtained by considering different fractional orders a , a , and a for either wave and damped wave models. These results show the 1 2 3 convergence, accuracy, wellposedness, and speeds of the method to deal with the high-dimensional wave and damped wave models well. To obtain the best value for the shape parameter, a trial and error process is used such that there is an equilibrium between the accuracy of the suggested computational procedure and obtained condition number of the coefficient matrix. Hence, the plot of absolute errors and condition numbers for different shape parameters by considering N = 9 and a = a = a = 1.50 are displayed in Figure 17 for both wave and damped wave models. 1 2 3 Furthermore, Table 9 indicates the temporal convergence of the suggested numerical method to deal with mentioned models. The results are obtained vs. N by considering N = 12 and shape t x parameter c = 0.060 for wave model and c = 0.060 for damped wave model. To investigate the method’s stability numerically, the impression of the input noise on the initial solution is evaluated. For this purpose, the effects of the several input noise levels s = 0, s = 0.001, s = 0.01, and s = 0.1 on error estimates are considered. These results are reported in Table 4 and the acceptable and stable behavior of the numerical procedure against the input noise are illustrated. The coefficient matrix sparsity patterns for both models are shown in Figure 18. Axioms 2021, 10, 259 20 of 22 Table 8. Error estimates, condition numbers, CPU times and convergence rate of Example 4 by letting n = 15 and different values of N and several fractional orders a , a , a . 1 2 3 Wave Model Damped Wave Model (a , a , a ) N 1 2 3 c e e CN Time R c e e CN Time R ¥ r ¥ ¥ r ¥ 4 4 5 3 5 5 3 (1.5, 1.5, 1.5) 5 0.030 3.3885 10 1.9901 10 1.9289 10 0.38 0.009 7.9669 10 1.4917 10 2.1248 10 0.389 4 5 6 4 5 6 4 7 0.035 3.4788 10 7.4475 10 6.5765 10 2.00 6.76 0.011 3.7856 10 7.4366 10 6.4930 10 2.02 2.21 4 5 6 5 5 6 5 9 0.074 1.8830 10 4.3033 10 5.5365 10 6.91 2.44 0.014 2.0860 10 4.2544 10 5.9485 10 7.01 2.37 4 5 6 6 5 6 6 11 0.099 1.1783 10 2.7394 10 3.0414 10 19.03 2.33 0.019 1.8689 10 3.0590 10 2.5062 10 19.42 0.54 4 5 5 3 5 6 3 (1.8, 1.7, 1.6) 5 0.030 6.5181 10 1.1376 10 2.5678 10 0.35 0.010 5.4917 10 9.5723 10 2.7288 10 0.36 4 5 6 4 5 6 4 7 0.037 2.5089 10 5.3738 10 2.6462 10 1.79 2.83 0.014 2.2968 10 4.7842 10 2.6368 10 1.81 2.59 4 5 6 5 5 6 5 9 0.041 1.4295 10 3.0710 10 6.9574 10 6.42 2.23 0.017 1.9604 10 3.3110 10 7.5516 10 6.55 0.63 4 6 6 6 6 6 6 11 0.050 9.4011 10 1.9582 10 9.1183 10 18.47 2.08 0.020 7.6139 10 1.5645 10 7.1603 10 20.01 4.71 4 5 5 3 5 6 3 (1.5, 1.9, 1.7) 5 0.020 6.0536 10 1.0547 10 2.7232 10 0.35 0.011 5.5235 10 9.6439 10 2.8963 10 0.36 4 5 6 4 5 6 4 7 0.030 2.5729 10 5.3827 10 2.6185 10 1.75 2.54 0.015 2.3448 10 4.8658 10 2.6109 10 1.90 2.54 4 5 6 5 5 6 5 9 0.035 1.6602 10 3.9895 10 4.9197 10 6.51 1.74 0.018 1.7139 10 3.8854 10 9.4595 10 7.23 1.24 4 5 6 6 6 6 6 11 0.035 1.1607 10 2.3294 10 4.3370 10 18.11 1.78 0.021 8.3998 10 1.7479 10 3.8080 10 22.24 3.55 Table 9. Error estimates for N = 12 of Example 4 by letting c = 0.060 for wave model and c = 0.025, n = 15, different values of N and several fractional orders a , a , a . c t 1 2 3 Wave Model Damped Wave Model (a , a , a ) N 1 2 3 t e e e e ¥ r ¥ r 4 4 4 4 (1.7, 1.7, 1.7) 3 9.2461 10 1.8013 10 9.4060 10 1.8327 10 4 4 4 4 6 8.7935 10 2.0139 10 9.1323 10 2.0482 10 5 6 5 6 9 1.4375 10 3.2206 10 1.6470 10 2.9951 10 5 6 6 6 12 1.2491 10 2.1729 10 5.6840 10 1.2599 10 4 4 4 4 (1.5, 1.7, 1.9) 3 8.9200 10 1.7432 10 9.0557 10 1.7705 10 4 4 4 4 6 8.4834 10 1.9489 10 8.7932 10 1.9787 10 5 6 5 6 9 2.0456 10 4.8243 10 1.5055 10 3.8044 10 6 6 6 6 12 9.2618 10 2.0153 10 7.6079 10 1.5858 10 4 4 4 4 (1.8, 1.7, 1.6) 3 9.1571 10 1.7857 10 9.3117 10 1.8162 10 4 4 4 4 6 8.7089 10 1.9965 10 9.0402 10 2.0297 10 5 6 5 6 9 1.9037 10 4.5341 10 2.2248 10 3.9412 10 5 6 6 6 12 1.1909 10 2.1102 10 6.2373 10 1.3494 10 (a) (b) Figure 17. Absolute errors and condition numbers for Example 4 by letting N = 9 and a = a = a = 1.50 for different shape parameter c with respect to (a) wave model and (b) damped 1 2 3 wave model. Axioms 2021, 10, 259 21 of 22 (a) (b) Figure 18. The sparsity pattern of the coefficient matrix for Example 4 with respect to (a) wave model (b) damped wave model. 4. Conclusions Local Mashless numerical methods have widely proved their ability to solve efficiently partial differential equations. In this work for solving the wave and damped wave equation with Riemann–Liouville fractional derivatives, we focus on the space–time Gaussian radial basis function in a high-dimensional setting. The proposed technique has some noticeable features, such as constructing a sparse matrix that reduces the time execution of the implementation of algorithms. This feature allows the method to be applied to high- dimensional problems that occur in nature and engineering as well as the using the RBFs makes it much easier to work on high-dimensional spaces with irregular computational domains. Moreover, employing the space–time Gaussian RBF eliminates the need for time discretization of the model. Numerical experiments verify that the local features lead to the sparse coefficient matrix and reduce the computational costs in high-dimensional problems. Improvements in terms of accuracy and efficiency on the computed solution have been confirmed by the proposed numerical scheme. Finally, we aim to apply this method to more complex fractional partial differential equations with applications in computational finance. Author Contributions: Conceptualization, M.R. and S.C.; methodology, M.R. and S.C.; software, M.R.; formal analysis, M.R. ans S.C. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: This work is partially supported by INdAM-GNCS, “Research ITalian network on Approximation (RITA)” and UMI Group TAA “Approximation Theory and Applications”. Conflicts of Interest: The authors declare no conflict of interest. References 1. Mainardi, F. Fractals and Fractional Calculus in Continuum Mechanics; Springer: Vienna, Austria, 1997; pp. 291–348. 2. Assari, P.; Cuomo, S. The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines. Eng. Comput. 2019, 35, 1391–1408. [CrossRef] 3. De Marchi, S.; Martinez, A.; Perracchione, E.; Rossini, M. RBF-based partition of unity methods for elliptic PDEs: Adaptivity and stability issues via variably scaled kernels. J. Sci. Comput. 2019, 79, 321–344. [CrossRef] 4. Ala, G.; Fasshauer, G.E.; Francomano, E.; Ganci, S.; McCourt, M.J. An augmented MFS approach for brain activity reconstruction. Math. Comput. Simul. 2017, 141, 3–15. [CrossRef] 5. Ala, G.; Francomano, E.; Fasshauer, G.E.; Ganci, S.; McCourt, M.J. A meshfree solver for the MEG forward problem. IEEE Trans. Magn. 2015, 51, 1–4. [CrossRef] 6. Wendland, H. Scattered Data Approximation (Cambridge Monographs on Applied and Computational Mathematics); Cambridge University Press: Cambridge, UK, 2004. Axioms 2021, 10, 259 22 of 22 7. Kansa, E.J. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput. Math. Appl. 1990, 19, 127–145. [CrossRef] 8. Kansa, E.J. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 1990, 19, 147–161. [CrossRef] 9. De Marchi, S.; Martínez, A.; Perracchione, E. Fast and stable rational RBF-based partition of unity interpolation. J. Comput. Appl. Math. 2019, 349, 331–343. [CrossRef] 10. Campagna, R.; Cuomo, S.; De Marchi, S.; Perracchione, E.; Severino, G. A stable meshfree PDE solver for source-type flows in Porous media. Appl. Numer. Math. 2020, 149, 30–42. [CrossRef] 11. Uhlmann, V.; Delgado-Gonzalo, R.; Conti, C.; Romani, L.; Unser, M. Exponential Hermite splines for the analysis of biomedical images. In Proceedings of the 2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy, 4–9 May 2014; pp. 1631–1634. 12. Kansa, E.; Hon, Y. Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations. Comput. Math. Appl. 2000, 39, 123–137. [CrossRef] 13. Ling, L.; Kansa, E.J. A least-squares preconditioner for radial basis functions collocation methods. Adv. Comput. Math. 2005, 23, 31–54. [CrossRef] 14. Wang, F.; Zheng, K.; Ahmad, I.; Ahmad, H. Gaussian radial basis functions method for linear and nonlinear convection–diffusion models in physical phenomena. Open Phys. 2021, 19, 69–76. [CrossRef] 15. Wendland, H. Fast evaluation of radial basis functions: Methods based on partition of unity. In Approximation Theory X: Wavelets, Splines, and Applications; Vanderbilt University Press: Neshville, TE, USA, 2002. 16. Cavoretto, R.; De Rossi, A. Adaptive meshless refinement schemes for RBF-PUM collocation. Appl. Math. Lett. 2019, 90, 131–138. [CrossRef] 17. Cavoretto, R.; De Rossi, A. Error indicators and refinement strategies for solving Poisson problems through a RBF partition of unity collocation scheme. Appl. Math. Comput. 2020, 369, 124824. [CrossRef] 18. Li, J.F.; Ahmad, I.; Ahmad, H.; Shah, D.; Chu, Y.M.; Thounthong, P.; Ayaz, M. Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method. Open Phys. 2020, 18, 1063–1072. [CrossRef] 19. Nawaz Khan, M.; Ahmad, I.; Akgül, A.; Ahmad, H.; Thounthong, P. Numerical solution of time-fractional coupled Korteweg–de Vries and Klein–Gordon equations by local meshless method. Pramana 2021, 95, 1–13. 20. De Marchi, S.; Wendland, H. On the convergence of the rescaled localized radial basis function method. Appl. Math. Lett. 2020, 99, 105996. [CrossRef] 21. Piret, C.; Hanert, E. A radial basis functions method for fractional diffusion equations. J. Comput. Phys. 2013, 238, 71–81. [CrossRef] 22. Myers, D.; De Iaco, S.; Posa, D.; De Cesare, L. Space–time radial basis functions. Comput. Math. Appl. 2002, 43, 539–549. [CrossRef] 23. Fasshauer, G.E. Meshfree Approximation Methods with MATLAB; World Scientific Publishing Co., Pte. Ltd.: Singapore, 2007.
http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png
Axioms
Multidisciplinary Digital Publishing Institute
http://www.deepdyve.com/lp/multidisciplinary-digital-publishing-institute/an-efficient-localized-meshless-method-based-on-the-space-time-B1Cbeup56h