Research paper
Shifted Jacobi–Gauss-collocation with convergence analysis for fractional integro-differential equations

https://doi.org/10.1016/j.cnsns.2019.01.005Get rights and content

Highlights

  • A new shifted Jacobi–Gauss-collocation algorithm is presented.

  • Different classes of fractional integro-differential equations are addressed.

  • Error analysis is performed.

  • Numerical examples are given for illustrating the method advantages.

Abstract

A new shifted Jacobi–Gauss-collocation (SJ-G-C) algorithm is presented for solving numerically several classes of fractional integro-differential equations (FI-DEs), namely Volterra, Fredholm and systems of Volterra FI-DEs, subject to initial and nonlocal boundary conditions. The new SJ-G-C method is also extended for calculating the solution of mixed Volterra–Fredholm FI-DEs. The shifted Jacobi–Gauss points are adopted for collocation nodes and the FI-DEs are reduced to systems of algebraic equations. Error analysis is performed and several numerical examples are given for illustrating the advantages of the new algorithm.

Introduction

Fractional differential equations (FDEs) [1], [2], [3], [4], [5] are powerful tools for modeling phenomena in mathematical chemistry [6], [7], biology [8], viscoelasticity [3], physics [4], and other areas [9], [10], [11], [12], [13], [14]. The increasing applicability of FDEs has required efficient algorithms for calculating their solutions. However, since most FDEs cannot be solved analytically, numerical methods have been developed [10], [11], [12]. Despite the intense research that has been carried out in this topic, the problem is still challenging.

Fractional integro-differential equations (FI-DEs) are widely used in science and engineering. For example, the activity of interacting inhibitory and excitatory neurons is well modeled by means of FI-DEs. Detailed numerical methods for solving one-dimensional FI-DEs were presented in [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], while several techniques for general FI-DEs were developed by many authors. For example, Nazari and Shahmorad [25] introduced the fractional differential transform method for FI-DEs with nonlocal boundary conditions. Jiang and Tian [26] improved the reproducing kernel scheme for nonlinear Volterra FI-DEs. Saeedi and Moghadam [27] used the CAS wavelets technique for solving nonlinear Volterra FI-DEs of arbitrary order. Susahab et al. [28] applied quadrature rules to a class of nonlinear FI-DEs of the Hammerstein type. Zhu and Fan [29] developed the second kind Chebyshev wavelet for solving nonlinear Fredholm FI-DEs. Other authors [30], [31], [32], [33], [34], [35] developed and employed different numerical techniques for solving FI-DEs.

Numerical methods are divided into local and global techniques. The finite-difference and finite-element methods are classified as local techniques, whilst the spectral method is global. In practice, finite-element methods are particularly well suited to problems in complex geometries, whereas spectral methods can provide superior accuracy, at the expense of domain flexibility. We emphasize that there are many numerical approaches, such as hp finite-elements and spectral-elements, which combine advantages of both the global and local methods. However in this paper, we shall restrict our attentions to the global spectral methods. These can be thought as a development of the so-called method of weighted residuals.

Recently, spectral methods were recognized as efficient numerical schemes for solving FI-DEs [32], [33]. Spectral methods are characterized by having faster convergence rates and better accuracy than the local methods. According to this method, the solution of FI-DEs is expressed in terms of a finite series of known functions, which are global in the sense that they are defined over the entire domain and are called trial/basis functions. After substituting this series in the FI-DEs, an inner product of the resulting equation with the so-called test functions is formed, which is used in order to guarantee that the equation is satisfied as closely as possible by the truncated series expansion. This is accomplished by minimizing the error in the differential equation produced by using the truncated series expansion instead of the exact solution, with respect to a suitable inner product. Regarding the methodology used, the spectral methods are divided into four categories, namely collocation [36], [37], [38], [39], [40], tau [41], [42], Galerkin [43] and Petrov-Galerkin [44] methods.

In this paper, we propose an accurate numerical algorithm for calculating the solutions of different classes of FI-DEs with initial and nonlocal boundary conditions. Using the shifted Jacobi–Gauss collocation (SJ-G-C) method with the Riemann–Liouville (R-L) fractional derivative of the shifted Jacobi polynomials, we reduce the FI-DEs to systems of algebraic equations. The solution of such equations is approximated by means of a finite expansion of shifted Jacobi polynomials for independent variables (for more details see Canuto et al. [45]). Then we evaluate the residuals of the mentioned problem at the shifted Jacobi–Gauss quadrature points. Substituting these approximations in the FI-DEs leads to a system of algebraic equations. This system may be solved numerically using the Newton’s iterative algorithm. This scheme is one of the most suitable methods for solving systems of algebraic equations. Indeed, with the freedom to select the shifted Jacobi indexes σ>1,ρ>1, the method can be calibrated for a wide variety of problems. Moreover, we develop and analyze spectral collocation methods based on Jacobi polynomials with general parameters σ and ρ. The main advantage of the proposed algorithm is that the Chebyshev, Legendre and ultraspherical collocation methods can be obtained as special cases from our method. Furthermore, an error analysis of the new method is developed and the results are discussed.

The paper is organized as follow. Section 2 introduces the tools of fractional calculus and the shifted Jacobi polynomials. Section 3 applies the new SJ-G-C to one-dimensional linear Volterra FI-DEs subject to nonlocal conditions, and to non-linear Volterra FI-DEs subject to initial conditions. Sections 4–6 extend the SJ-G-C method to solve one-dimensional non-linear Fredholm FI-DEs, systems of Volterra FI-DEs, and mixed Volterra–Fredholm FI-DEs, respectively. Section 7 presents some useful lemmas and error analysis. Section 8 solves some numerical examples and, finally, Section 9 draws the main conclusions.

Section snippets

Fractional calculus

The fractional integral and derivative of order ν > 0 can be expressed by means different formulas. Often we use the R-L definitions.

Definition 2.1

The R-L fractional integral of order ν > 0, Jν, is given byJν(z)=1Γ(ν)0z(zζ)ν1(ζ)dζ,ν>0,z>0,J0(z)=(z), whereΓ(ν)=0zν1ezdz.The operator Jν satisfiesJνJμ(z)=Jν+μ(z),JνJμ(z)=JμJν(z),Jνzρ=Γ(ρ+1)Γ(ρ+1+ν)zρ+ν.

Definition 2.2

The R-L fractional derivative of order ν > 0, Dν, is given byDν(z)=1Γ(mν)dmdzm(0z(zζ)mν1(ζ)dζ),m1<νm,z>0,where m is the ceiling function of

Linear Volterra FI-DEs with nonlocal boundary condition

The SJ-G-C method is applied to numerically solve the linear Volterra FI-DE with nonlocal conditionsDν(z)=f(z)+0zk(z,ζ)(ζ)dζ,0<ν<1,(0)+γ(1)+λabϕ(ζ)(ζ)dζ=d1,where f(z), ϕ(ζ) and k(z, ζ) are given functions, γ and λ are constants, and ℜ(z) is an unknown function.

The solution of Eq. (3.1) is approximated byN(z)=j=0NajPL,j(σ,ρ)(z),and the fractional derivative of ℜN(z) is estimated asDνN(z)=j=0NajDν(PL,j(σ,ρ)(z)).Given the R-L derivativeDνzk=1Γ(1ν)(0zχk(zχ)νdχ),=zkνΓ(1+k)Γ(1+kν),0<ν

Non-linear Fredholm FI-DEs with initial conditions

The SJ-G-C method is applied to numerically solve the Fredholm FI-DE with initial conditionsDν(z)=f(z)+0Lk(z,ζ)((ζ))pdζ,1<ν<2,subject tou(m)(0)=dmm=0,1.The solution of Eq. (4.1) is approximated byN(z)=j=0NajPL,j(σ,ρ)(z).Based on the results in the last subsections, we obtainj=0NajΨL,j(σ,ρ)(z)=f(z)+0L(k(z,ζ)j=0NajPL,j(σ,ρ)(ζ))dζ,thus, we getj=0Naj(ΨL,j(σ,ρ)(zL,N,i(σ,ρ))0Lk(zL,N,i(σ,ρ),t)PL,j(σ,ρ)(ζ)dζ)=f(zL,N,i(σ,ρ))i=1,,N1.Merging Eqs. (4.3) and (4.2), yieldsj=0NajDmPL,j(σ,ρ)(0)=dm

System of Volterra FI-DEs with initial conditions

In this Section, we extend the SJ-G-C technique to solve the system of Volterra FI-DEs{Dν(z)=v(z)+f(z)+0zk1(z,ζ)[(ζ)+v(ζ)]dζ,0<ν<1,Dνv(z)=(z)+g(z)+0zk2(z,ζ)[(ζ)+v(ζ)]dζ, subject to the conditions(0)=d1,v(0)=d2.Here, we approximate ℜ(z) and v(z) using shifted Jacobi polynomialsN(z)=j=0NajPL,j(σ,ρ)(z),vN(z)=j=0NbjPL,j(σ,ρ)(z).Using (5.3), we deduce that{j=0NajDνPL,j(σ,ρ)(z)=j=0NbjPL,j(σ,ρ)(z)+f(z)+0z(k1(z,ζ)[j=0NajPL,j(σ,ρ)(ζ)+j=0NbjPL,j(σ,ρ)(ζ)])dζ,j=0NbjDνPL,j(σ,ρ)(z)=j=0NajPL,j

Mixed Volterra–Fredholm FI-DEs with nonlocal boundary conditions

We present the SJ-G-C method to numerically solve the linear fractional mixed Volterra–Fredholm FI-DEDν(z)=f(z)+0zk(z,ζ)(ζ)dζ+0Lk(z,ζ)(ζ)dζ, with the nonlocal boundary conditions(0)+γ(1)+λabϕ(ζ)(ζ)dζ=d1.Based on the results presented in the previous subsections, we obtain the following system of algebraic equationsj=0Naj(PL,j(σ,ρ)(0)+γPL,j(σ,ρ)(1)+λ(abϕ(ζ)PL,j(σ,ρ)(ζ)))=d1,j=0Naj(ΨL,j(σ,ρ)(zL,N,i(σ,ρ))0zL,N,i(σ,ρ)k(zL,N,i(σ,ρ),ζ)PL,j(σ,ρ)(ζ)dζ0Lk(zL,N,i(σ,ρ),ζ)PL,j(σ,ρ)(ζ)dζ)=f(z

Lemmas and error analysis

In this Section useful lemmas and error analysis of the J-G-C algorithm presented in Section 3.1.

Numerical results

This Section presents several examples to illustrate the accuracy and effectiveness of the proposed SJ-G-C method.

We define the absolute error, E(z), asE(z)=(z)N(z), where ℜ(z) and ℜN(z) are the exact and the approximate solutions at point z, respectively. The maximum absolute error, MAE, is given byMAE=max{E(z)}.

Moreover, we define the root mean square error, RMSE, asem(z)2=(i=0N((zL,N,i(σ,ρ))N(zL,N,i(σ,ρ)))2)12N+1.

Example 1

Firstly, we introduce the linear Volterra FI-DE [25]D13(z)=32z23Γ(23

Conclusion

In this paper efficient numerical techniques based on the SJ-G-C method were developed for solving FI-DEs subject to initial and nonlocal conditions. The SG-G points were adopted for collocation nodes and the FI-DEs reduced to systems of algebraic equations. Spectral methods are promising candidates for solving many problems, since their global nature fits well with the nonlocal definition of fractional operators. Spectral methods can be used to solve linear and nonlinear differential

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      Doha et. al [94] used Shifted JacobiGauss-collocation method for solving these problems. Recently, analysing the convergence of the Chebyshev Legendre spectral method in solving Fredholm fractional integro-differential equations was investigated in [95] and using Lucas wavelets (LWs) and the Legendre Gauss quadrature rule to solve the fractional FredholmVolterra integro-differential equations was presented in [96].

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