Numerical solution of time-fractional fourth-order reaction-diffusion model arising in composite environments
Introduction
The idea behind fractional calculus (FC) has a long history aligned with that of the integer or ordinary calculus [1], [2], [3], [4]. The FC generalizes the integer order integrals and derivatives to real or even variable orders. The FC can also be interpreted of as a field of mathematical analysis that investigates differential operators and equations where the integrals are of convolution or weakly singular type. In the past, the FC was regarded as an abstract theory, but recently a diversity of areas of science and engineering, including biology, biophysics, physics, chemistry, and control theory, adopted successfully fractional order differential equations [5], [6], [7], [8], [9].
The fourth-order reaction-diffusion equations play a crucial role in significant fields, such as pattern formation of bistable systems, propagation of domain walls in liquid crystals, traveling waves of reaction-diffusion systems, thin beams, brain warpin, strain gradient elasticity, and fluids on lung, and others (see [10], [11], [12], [13], [14], [15]). In the current work, our goal is to investigate the approximated solution of the time-fractional fourth-order reaction-diffusion equation (TF4RDE) that can be written assubject to the following general initial and boundary conditionswhere Δ and Δ2 represent the Laplacian and bi-Laplacian operators respectively with respect to the spatial variables, 0 < β < 1, γ is the positive constant, Ω denotes a bounded convex polygonal space domain in ∂Ω stands for the boundary of Ω, and f(x, t) is a sufficiently smooth function. The fractional derivative ∂βu(x, t)/∂tβ in the Riemann-Liouville fractional derivative of order β is written aswith Γ( · ) denoting the Gamma function [1], [2], [3], [4].
In spite of its importance we find only a few numerical algorithms for solving the TF4RDE. Du et al. [16] presented a local discontinuous Galerkin method. They introduced the auxiliary variables to reduce the TF4RDE into the first-order system. Wei et al. [17] adopted a mixed finite element (FE) scheme. Liu et al. [18] developed a two-grid mixed FE technique. Abbaszadeh and Dehghan [19] used the collocation method based on the direct meshless local Petrov–Galerkin (DMLPG) method for approximating the TF4RDE. Remark 1 Following the discussion [17], [18], [20], [21], the TF4RDE can be written as a coupled system of two lower-order equations by introducing an auxiliary variable to reduce the impact of fourth-order derivative,
The Riemann-Liouville and the Caputo fractional derivatives are often selected for time fractional derivatives. The following relationship is generally considered between these two fractional operators [1], [2], [3], [4]: where and denote the Riemann-Liouville and Caputo fractional derivative, respectively.
The Riemann-Liouville fractional derivative in (5) is significant since an excellent opportunity to weaken the conditions corresponding to the function u(t) is provided by (4). One needs to use these weak conditions, for example, to reach the solution for the Abel integral equation (the reader can refer to page 62 in the reference [2] for a comprehensive treatment). The Riemann-Liouville fractional derivative (4) is selected here according to [22], [23]. Lin et al. [22] and Zhang et al. [23] investigated various numerical techniques for Riemann-Liouville fractional partial differential equations and followed (5) in order to provide a comprehensive numerical analysis. It is possible to adopt numerical techniques for the Riemann-Liouville time fractional derivative without using the relationship (5), as demonstrated in [24]. The Riemann-Liouville fractional differential equations can be solved by considering the initial value term in various forms. These forms include the integral formulation obtained by applying the Laplace transform [25] and the one considered in the main problem (1) that is obtained from numerical approximations [22], [23], [26].
Most of the classical numerical algorithms, such as the finite element (FE), finite difference (FD), finite volume (FV), and pseudo-spectral (PS) methods, that are used to approximate the solutions, require the data arrangement in a structured pattern. They also ask for data confined to regions having a simple shape, such as a circle or a rectangle. However, in many applications, it is difficult to satisfy this strict requirement and, therefore, classical numerical techniques may not be applicable. One popular method for solving these problems is the meshfree (meshless) scheme. The main benefits of the methods based on meshfree can be briefly outlined as [27]: Meshfree techniques (i) do not require any a priori information on the mesh, and there is no need to generate a mesh throughout the problem solving process, (ii) have a straightforward implementation in high dimensions on a set of uniform or scattered data points in regular and irregular geometries and do not need a mesh, (iii) represent an economical alternative to classical techniques in order to eliminate their structural requirements.
The radial basis function (RBF) methods are grid-independent and, hence, fall into a class of meshfree approaches used to interpolate multidimensional scattered data. With this technique, the approximation is dependent on the inter-nodal distances and is truly meshfree since it does not require any points or meshes to be connected. Determining the inter-nodal distances in RBF poses usually a lower computational burden than the mesh generation in mesh-based techniques. Particularly for a large number of dimensions, generating a mesh in mesh-based techniques is laborious, but the difficulty of calculating the inter-nodal distances in RBF does not increase. The multiquadric (MQ) RBF was introduced in 1968 by Hardy and we find a detailed discussion about the scheme in [28]. Later, Hardy reported the development of MQ from 1968 to 1988 [29]. The MQ interpolation received little attention until 1979, when Franke [30], [31] demonstrated that the MQ RBF is the most suitable technique for problems of scattered data interpolation. Moreover, he also speculated that the system matrix in this technique is invertible and that it is a well-posed technique. Indeed, no theoretical formulation of the method was available at that time, but later Micchelli [32] demonstrated that the system matrix of the MQ, as well as those of numerous other RBF techniques, are invertible. Kansa [33], [34] was the first to apply MQ RBF for solving PDE. Despite their good accuracy, RBF techniques are ill-conditioned. This relationship manifests itself in the RBF interpolation as the uncertainty principle, first documented by Schaback [35]. In the case of elliptic problems, Larson and Fornberg [36] noted that RBF-based algorithms are more accurate than those using the Fourier–Chebyshev PS and the second-order FD techniques. Madych et al. [37] and Buhmann et al. [38] showed that infinitely smooth RBF produces a spectral convergence order that is faster than any polynomial order. Other works about the uniqueness, existence, and convergence of the RBF interpolation were discussed in [32], [37], [39].
Throughout this article, we suppose that the considered problem has is a unique solution with sufficiently smooth properties as defined in Ω × [0, T] for the problem (1)-(3) under consideration. Whether the problem is solvable can be determined via the Leray–Schauder fixed point theorem used in [40], [41]. In addition, f(x, t) and u(x, 0) are smooth enough to satisfy the requirements of the present theoretical analysis. Furthermore, we assume that is continuous over Ω × [0, T]. Due to the lack of smoothness in the solution near the starting time (), one cannot hold the regularity assumption of u ∈ C3[0, T] with respect to time. As a result, one can make some appropriate regularity assumptions by referring to [42], [43], [44], [45], [46], [47], [48]and, particularly, the detailed analysis about the regularity of solution as in [42], [43], [48], [49].
This paper introduces a new hybrid method based RBF for obtaining the solution of the TF4RDE. The paper is divided into five sections and is organized as follows. Section 2 implements a finite difference approach with second-order accuracy for discretizing the problem in the time direction. Moreover, a detailed proof of stability of the time-discretize approach and error estimate are given. Section 3 develops a new technique denoted as LRBF-FD (local RBF-generated finite difference) to discretize the spatial derivative for every point in the corresponding local-support domain. Section 4 provides some numerical results that clarify the theoretical analysis on the simple and complex domains with the irregular and non-regular grid points. Finally, Section 5 presents the conclusions.
Section snippets
The time fractional discretization of the problem
For discretization of time variable, we require some preliminary. Let be the time step size and the time grid points for Moreover, for a smooth function ψ on [0, T], let us define . In what follows, we require the following lemmas to formulate the semi-discretization in time following Refs [24], [26], [50]. Lemma 1 For approximating at time we get the discrete formulawhere
The formulation of the LRBF-FD method
In the RBF interpolation, the approximate solution of a PDE can be represented by a linear combination of smooth RBF at set of centers where are unknown coefficients, are RBF with shape parameter ε, and the norm ‖ · ‖ is the Euclidean norm [38], [62]. The coefficients, are chosen by enforcing the interpolation condition at a set of nodes that typically coincides with the centers. The
Numerical tests and discussion
This section analyses the accuracy and efficiency of the new method using three numerical examples and evaluates the impact of several values of δt and h. For a weak regularity solution u, the time convergence order is lower than the theoretical formulation that is expressed in Example 4.2. The convergence order in the temporal and spatial variables are calculated by the following relationsrespectively, where
Concluding remarks
The reaction-diffusion equations explain a variety of biological, chemical, and physical phenomena that arise in composite environments. This paper considered a numerical meshless method based on the LRBF-FD to achieve the approximation solution of the TF4RDE. The new method is particularly adequate for arbitrarily distributed nodes and complex problem domains, where the conventional approaches are pose considerable limitations. The new technique consists of two phases. First, the second-order
Acknowledgements
The authors would like to thank the anonymous referees and editor for their valuable comments and suggestions on the revision of this article. In addition, the authors are deeply indebted to the respected eminent Professor Zakieh Avazzadeh from Department of Mathematical Science, Xi’an Jiaotong-Liverpool University Suzhou 215123, Jiangsu, China, for her carefully reading the revised version and suggestions on how to improve this work.
References (76)
- et al.
Finite difference/Hermite–Galerkin spectral method for multi-dimensional time-fractional nonlinear reaction–diffusion equation in unbounded domains
Applied Mathematical Modelling
(2019) - et al.
A meshless method for solving three-dimensional time fractional diffusion equation with variable-order derivatives
Applied Mathematical Modelling
(2020) - et al.
Numerical solution of multi-order fractional differential equations with multiple delays via spectral collocation methods
Applied Mathematical Modelling
(2018) - et al.
Legendre wavelets approach for numerical solutions of distributed order fractional differential equations
Applied Mathematical Modelling
(2019) - et al.
A mathematical model for atmospheric ice accretion and water flow on a cold surface
International Journal of Heat and Mass Transfer
(2004) - et al.
Implicit brain imaging
NeuroImage
(2004) - et al.
Local discontinuous Galerkin method for a nonlinear time-fractional fourth-order partial differential equation
Journal of Computational Physics
(2017) - et al.
A mixed finite element method for a time-fractional fourth-order partial differential equation
Applied Mathematics and Computation
(2014) - et al.
A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative
Computers & Mathematics with Applications
(2015) - et al.
Direct meshless local Petrov–Galerkin (DMLPG) method for time-fractional fourth-order reaction–diffusion problem on complex domains
Computers & Mathematics with Applications
(2020)
Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem
Computers & Mathematics with Applications
Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems
Applied Mathematical Modelling
Finite element method combined with second-order time discrete scheme for nonlinear fractional Cable equation
The European Physical Journal Plus
Experimental evidence for fractional time evolution in glass forming materials
Chemical physics
Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term
Computers & Mathematics with Applications
Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988
Computers & Mathematics with Applications
Recent advances in the approximation of surfaces from scattered data
Topics in multivariate approximation
Multiquadrics ”A scattered data approximation scheme with applications to computational fluid-dynamics” I surface approximations and partial derivative estimates
Computers & Mathematics with applications
Multiquadrics–a scattered data approximation scheme with applications to computational fluid-dynamics–II solutions to parabolic, hyperbolic and elliptic partial differential equations
Computers & mathematics with applications
A numerical study of some radial basis function based solution methods for elliptic PDEs
Computers & Mathematics with Applications
Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator
Applied Mathematics and Computation
An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions
Applied Numerical Mathematics
A finite difference scheme for partial integro-differential equations with a weakly singular kernel
Applied Numerical Mathematics
Alternating direction implicit-euler method for the two-dimensional fractional evolution equation
Journal of Computational Physics
Error analysis of a finite element method with GMMP temporal discretisation for a time-fractional diffusion equation
Computers & Mathematics with Applications
Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence
Journal of Computational Physics
Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation
Journal of Computational Physics
Finite difference/spectral approximations for the time-fractional diffusion equation
Journal of Computational Physics
A fully discrete difference scheme for a diffusion-wave system
Applied Numerical Mathematics
Second-order approximation scheme combined with H1-Galerkin MFE method for nonlinear time fractional convection–diffusion equation
Computers & Mathematics with Applications
Meshfree explicit local radial basis function collocation method for diffusion problems
Computers & Mathematics with applications
An upwind local RBF-DQ method for simulation of inviscid compressible flows
Computer Methods in Applied Mechanics and Engineering
On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy
Journal of Computational Physics
On the role of polynomials in BF-FD approximations: II. Numerical solution of elliptic PDEs
Journal of Computational Physics
Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations
Computer Methods in Applied Mechanics and Engineering
Scattered node compact finite difference-type formulas generated from radial basis functions
Journal of Computational Physics
Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media
International Communications in Heat and Mass Transfer
Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation
Wave Motion
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