Abstract
By coupling of radial kernels and localized Laplace transform, a numerical scheme for the approximation of time fractional anomalous subdiffusion problems is presented. The fractional order operators are well suited to handle by Laplace transform and radial kernels are also built for high dimensions. The numerical computations of inverse Laplace transform are carried out by contour integration technique. The computation can be done in parallel and no time sensitivity is involved in approximating the time fractional operator as contrary to finite differences. The proposed numerical scheme is stable and accurate.
1. Introduction
In the last decades, many researchers have studied the fractional calculus [1–3]. Differential equations of fractional order have many applications in the field of science and engineering [4–7]. Analytical solution of many fractional differential equations is not possible or very hard to find, so we need a new numerical technique to find its approximate solution. Various phenomena in viscoelastic materials, economics, chemistry, finance, control theory, hydrology, physics, cosmology, solid mechanics, bioengineering, statistical mechanics, and control theory can be mathematically modeled from fractional calculus [8–17]. In literature, various numerical approaches are available for modeling anomalous diffusive behavior such as Carlo simulations [18]. An introduction of diffusion equations can be found in [19–21].
Recently, RBF-based methods were used in solving fractional partial differential equations (FPDEs) [22–24]. These methods have been employed in approximation of partial differential equations with complex domains. An implicit meshless technique based on the radial basis functions for the numerical simulation of the anomalous subdiffusion equation can be found in [25]. The convergence and stability of these mesh-free methods can be found in [26, 27]. These globally defined RBF methods cause ill-condition system matrices [28]. To overcome the problem of ill-conditioning, local RBF techniques were used in [29–31]. Unlike global RBF methods, the RBF method in local setting uses center points in each subdomain area of influence, surrounding each spatial point due to which there is reduction in the computational cost.
Recently, Laplace transform is combined with RBF method in [32, 33]. In [34–37], the authors use Laplace transform as tool in spectral method and other mesh-based methods such as finite element methods and finite difference method. To avoid the issues of computational efficiency and instability of the system matrix, we introduce a new technique Laplace transform-based local RBF method in solving the time fractional modified anomalous subdiffusion equations in irregular domain.
Here, we consider the following modified anomalous subdiffusion equation of fractional order [38]:where , , , subject to the following boundary and initial conditions:respectively, where , , , , are positive constants, is the Laplace operator, and is some given function.
2. Preliminaries
Here, we introduce some fundamental definitions related to fractional calculus [39, 40].
Definition 1. Let and , then the Caputo derivative of fractional order is defined as
Definition 2. Let , , be a given function, then its Laplace transform is defined byprovided this integral converges.
Lemma 1. If , with , then the Laplace transform of the fractional order Caputo derivative is given by
Theorem 1. the Bromwich inversion theorem [41]). Let w(t) have a continuous derivative and let , where and are positive constants. Definethen
3. Description of the Method
3.1. Time Discretization
Here, we apply Laplace transform to models (1)–(3) which gives
In more compact form, we have
The transformed problems (10) and (11) will be solved for the solution using local RBF method. The solution of the given models (1)–(3) will be found by using numerical inversion.
3.2. Local Radial Basis Functions Method
Here, the linear operators and are discretized by using local RBF [42, 43]. Consider the centers , where is the bounded domain. For each point , we can find a subdomain such that . The unknown function can be approximated with RBF in each local subdomain , , by the following equation:where are the unknown coefficients, and is the norm between nodes and , , is a radial kernel (multiquadric radial basis function), and is a local domain for around each , containing neighboring nodes around the node . So, we have small size linear systems each of order given bywhich can be denoted bywhere , and matrix is the system matrix.
Now, applying the operator to (12) gives
The vector form of (15) is given bywhere is given by
From equation (14), the unknown coefficients are given byand by inserting the values of in (16), we havewhere
Hence, the discretized form is given bywhere matrix is called the sparse differentiation matrix of order .
4. Numerical Inversion Technique
In this section, the numerical inversion of Laplace transform for approximating the given models (1)–(3) is as follows:where is the suitable path joining to . This Bromwich integral is numerically solved by using the following hyperbolic contour [37]:with , , , and .
Integral in (22) gives
Next applying trapezoidal rule for approximation of (24), we havewhere is the step size.
5. Application of the Method
In this section, the proposed numerical scheme is applied to multidimensional problems. We solved four test problems and used various domain points , stencils points , and quadrature points . Three error formulas, the error estimate, , , and norms are used. The radial kernel used in our computations is . The shape parameter is optimized by the uncertainty rule related to RBFs.
Problem 1. Consider models (1)–(3) to the following form [38]:with the following boundary and initial conditions:respectively, where the actual solution is given byIn our numerical scheme, we used the hyperbolic contour (23). The optimal parameter values are taken as, , , , , , and . This test problem is solved in the domain . Here, the number of points in domain is denoted by , the points in local subdomain are denoted by , and the number of quadrature points relates to . The numerical solutions are shown in Table 1 with various values of fractional order and and nodal points . For comparatively smaller values of fractional order and , better results in terms of and error norms are obtained. In the upper part of Table 1, condition number increases, as we increase nodal points . Error versus various quadrature points at , and and various values of and are shown in Figure 1. The error estimate for is well matched with and error norms, as shown in Figure 1. Hence, our proposed method is stable and accurate.
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Problem 2. Consider models (1)–(3) corresponding to the form [38]initial and boundary conditions given byThe actual solution isThe same domain and same parameter values as used in Problem 1 are incorporated. The numerical results are shown in Table 2 with the same as well as with various values of fractional order and and nodal points . For comparatively identical values of fractional order and , better results in terms of and error norms are obtained. In the upper part of Table 2, condition number of the system matrix is fixed for . Error versus various quadrature points at , and and various values of and are depicted in Figure 2. The error estimate for is well agreed with and error norms, as shown in Figure 1. The results obtained by our proposed numerical scheme are comparatively identical with the results in Table 2 [38].
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Problem 3. Next, we consider models (1)–(3) corresponding to the form [44]whereinitial and boundary conditions given byThe exact solution isThis problem is solved over the domain . In Table 3, for various nodal points and stencils points and with various values of and , the error norm is well matched with error norm. The condition number is increasing steadily as we decrease both the values of and at the same time.
Problem 4. Finally, we consider models (1)–(3) corresponding to the form [38]whereThe exact solution isHere, the problem is solved over the domain . In the upper section of Table 4, the and error norms are decreasing with , , , and and for nodal points . In the lower section of Table 4, for same values of and , the and error norms are decreasing steadily at , , and . The results are comparatively identical with the results of the paper [38]. Figure 3 shows the error with varying quadrature points and various values of and at , , and . The error is well matched with estimate for and error norm, as shown in Figure 3. The present method is stable and accurate in multidimensional fractional order partial differential equations.
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6. Conclusion
In this work, a numerical scheme is constructed which is based on Laplace transform and radial basis functions in the local setting. The proposed numerical scheme efficiently approximated time fractional anomalous subdiffusion equation. The supremacy of this method particularly for fractional order equations is its nonsensitive nature in time as contrary to finite difference approximation for fractional order operators. Since the fractional order derivative is of integral convolution type and suited to handle by Laplace transform, the spatial operators in multidimensions can be approximated by RBF in the local setting which generates small size differentiation matrices in local subdomains and these are assembled as a single sparse matrix in the global domain. So, large amount of data can be manipulated very easily and accurately.
Data Availability
The data supporting the results are available within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.