Numerical solution of time-fractional fourth-order reaction-diffusion model arising in composite environments

https://doi.org/10.1016/j.apm.2020.07.021Get rights and content

Highlights

  • The fractional fourth-order reaction-diffusion process is generated.

  • A new hybrid scheme based LRBF-FD method is formulated to approximate the time-fractional fourth-order reaction-diffusion model.

  • The LRBF-FD method useful for irregular domains with acceptable accuracy was proposed.

  • The stability and convergence of the proposed method are analyzed using the energy method.

Abstract

The fractional reaction-diffusion equation has an important physical and theoretical meaning, but its analytical solution poses considerable problems. This paper develops an efficient numerical process, the local radial basis function generated by the finite difference (named LRBF-FD) method, for finding the approximation solution of the time-fractional fourth-order reaction-diffusion equation in the sense of the Riemann-Liouville derivative. The time fractional derivative is approximated using the second-order accurate formulation, while the spatial terms are discretized by means of the LRB-FFD method. The advantage of the local collocation method is to approximate the differential operators by a weighted sum of the values of the function on a local set of nodes (local support) by deriving the RBF expansion. Furthermore, the ill-conditioned matrix resulting from using the global collocation method is avoided. The unconditional stability property and convergence analysis of the time-discrete approach are thoroughly proven and verified numerically. Three numerical examples are presented confirming the theoretical formulation and effectiveness of the new approach.

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 asu(x,t)t+βu(x,t)tβΔu(x,t)+Δ2u(x,t)+γu(x,t)=f(x,t),x=(x,y)ΩR2,0<tT,subject to the following general initial and boundary conditionsu(x,0)=g(x),xΩ¯=ΩΩ,u(x,t)=Δu(x,t)=0,xΩ,t>0,where 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 R2,∂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 asβu(x,t)tβ=0RLDtβu(x,t)=1Γ(1β)t0tu(x,η)(tη)βdη,with 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 σ=Δuto reduce the impact of fourth-order derivative,u(x,t)t+βu(x,t)tβΔu(x,t)+Δσ(x,t)+γu(x,t)=f(x,t),Δuσ=0.

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]:0RLDtβu(t)=0CDtβu(t)+u(0)tβΓ(β), where 0RLDtβu(t)and 0CDtβu(t)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 2ut2is continuous over Ω × [0, T]. Due to the lack of smoothness in the solution near the starting time (t=0), 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 δt=T/Mbe the time step size and the time grid points tk=kδtfor k=0,1,2,,M.Moreover, for a smooth function ψ on [0, T], let us define ψk=ψ(tk). In what follows, we require the following lemmas to formulate the semi-discretization in time following Refs [24], [26], [50].

Lemma 1

For approximating u(x,t)tat time tkβ2,we get the discrete formulau(x,tkβ2)t={Ptβukβ2+O(δt2),k2,Ptu1+O(δt),k=1,where Ptβukβ2=(3β)uk(42β)uk1+(1β)uk

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 XC={x1c,,xNc}Rd,u(x)S(x)=j=1Nαjϕj(x,ε), where {αj}j=1Nare unknown coefficients, ϕj(x,ε)=ϕ(xxjc2,ε),j=1,,N,are RBF with shape parameter ε, and the norm ‖ · ‖ is the Euclidean norm [38], [62]. The coefficients, {αj}j=1N,are chosen by enforcing the interpolation condition S(xic)=uic,i=1,,N,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 relationsCδt=log2(L(2δt,h)L(δt,h)),Ch=log2(L(4δt,2h)L(δt,h)),respectively, where L=max1jN1|U(xj,T)u

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)

  • Y. Liu et al.

    Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction–diffusion problem

    Computers & Mathematics with Applications

    (2015)
  • L. Wei et al.

    Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems

    Applied Mathematical Modelling

    (2014)
  • Y. Wang et al.

    Finite element method combined with second-order time discrete scheme for nonlinear fractional Cable equation

    The European Physical Journal Plus

    (2016)
  • R. Hilfer

    Experimental evidence for fractional time evolution in glass forming materials

    Chemical physics

    (2002)
  • N. Liu et al.

    Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term

    Computers & Mathematics with Applications

    (2018)
  • R.L. Hardy

    Theory and applications of the multiquadric-biharmonic method 20 years of discovery 1968–1988

    Computers & Mathematics with Applications

    (1990)
  • R. Franke

    Recent advances in the approximation of surfaces from scattered data

    Topics in multivariate approximation

    (1987)
  • E.J. Kansa

    Multiquadrics ”A scattered data approximation scheme with applications to computational fluid-dynamics” I surface approximations and partial derivative estimates

    Computers & Mathematics with applications

    (1990)
  • E.J. Kansa

    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

    (1990)
  • E. Larsson et al.

    A numerical study of some radial basis function based solution methods for elliptic PDEs

    Computers & Mathematics with Applications

    (2003)
  • T.-c. Wang et al.

    Analysis of some new conservative schemes for nonlinear Schrödinger equation with wave operator

    Applied Mathematics and Computation

    (2006)
  • H. Chen et al.

    An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions

    Applied Numerical Mathematics

    (2019)
  • T. Tang

    A finite difference scheme for partial integro-differential equations with a weakly singular kernel

    Applied Numerical Mathematics

    (1993)
  • L. Li et al.

    Alternating direction implicit-euler method for the two-dimensional fractional evolution equation

    Journal of Computational Physics

    (2013)
  • C. Huang et al.

    Error analysis of a finite element method with GMMP temporal discretisation for a time-fractional diffusion equation

    Computers & Mathematics with Applications

    (2020)
  • G.-H. Gao et al.

    Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence

    Journal of Computational Physics

    (2015)
  • Z. Wang et al.

    Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation

    Journal of Computational Physics

    (2014)
  • Y. Lin et al.

    Finite difference/spectral approximations for the time-fractional diffusion equation

    Journal of Computational Physics

    (2007)
  • Z.-z. Sun et al.

    A fully discrete difference scheme for a diffusion-wave system

    Applied Numerical Mathematics

    (2006)
  • J. Wang et al.

    Second-order approximation scheme combined with H1-Galerkin MFE method for nonlinear time fractional convection–diffusion equation

    Computers & Mathematics with Applications

    (2017)
  • B. Šarler et al.

    Meshfree explicit local radial basis function collocation method for diffusion problems

    Computers & Mathematics with applications

    (2006)
  • C. Shu et al.

    An upwind local RBF-DQ method for simulation of inviscid compressible flows

    Computer Methods in Applied Mechanics and Engineering

    (2005)
  • N. Flyer et al.

    On the role of polynomials in RBF-FD approximations: I. Interpolation and accuracy

    Journal of Computational Physics

    (2016)
  • V. Bayona et al.

    On the role of polynomials in BF-FD approximations: II. Numerical solution of elliptic PDEs

    Journal of Computational Physics

    (2017)
  • C. Shu et al.

    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

    (2003)
  • G.B. Wright et al.

    Scattered node compact finite difference-type formulas generated from radial basis functions

    Journal of Computational Physics

    (2006)
  • O. Nikan et al.

    Numerical approach for modeling fractal mobile/immobile transport model in porous and fractured media

    International Communications in Heat and Mass Transfer

    (2020)
  • J. Rashidinia et al.

    Numerical methods based on radial basis function-generated finite difference (RBF-FD) for solution of GKdVB equation

    Wave Motion

    (2019)
  • Cited by (42)

    • Localized meshless approaches based on theta method and BDF2 for nonlinear Sobolev equation arising from fluid dynamics

      2023, Communications in Nonlinear Science and Numerical Simulation
      Citation Excerpt :

      The spectral convergence rate of the MQ approximation was derived by Nelson and Madych [26]. Furthermore, the meshless RBF approaches have been widely applied in various fields recently and have become a feasible alternative in the scope of numerical analysis of PDEs [13,27–34]. This paper concentrates on an effective computational method by using the L-RBF-PUCM for approximating the nonlinear SE.

    • A suitable hybrid meshless method for the numerical solution of time-fractional fourth-order reaction–diffusion model in the multi-dimensional case

      2022, Engineering Analysis with Boundary Elements
      Citation Excerpt :

      Also, in 3D case we solve the time-fractional fourth-order reaction–diffusion model in the cylindrical, spherical and cubic domains. Moreover, a comparison between the presented scheme and the method in Ref. [59] is given in this section. Finally, a brief conclusion is given in Section 5.

    • A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms

      2022, Results in Physics
      Citation Excerpt :

      For solving time-FDWEs with the response and damping terms, Khalid et al. [52] developed the third-level modified B-spline functions. Other numerical approaches for treating time-FDWEs have been proposed, as may be seen in [53–55] and the references therein. The rest of this paper is structured as follows: the derivation of space derivatives through UCBS functions was explored in part 2.

    View all citing articles on Scopus
    View full text