Legendre wavelets method for solving fractional integro-differential equations

In this paper, based on the constructed Legendre wavelets operational matrix of integration of fractional order, a numerical method for solving linear and nonlinear fractional integro-differential equations is proposed. By using the operational matrix, the linear and nonlinear fractional integro-differential equations are reduced to a system of algebraic equations which are solved through known numerical algorithms. The upper bound of the error of the Legendre wavelets expansion is investigated in Theorem 5.1. Finally, four numerical examples are shown to illustrate the efficiency and accuracy of the approach.     

Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel

An integro-differential equation involving a convolution integral with a weakly singular kernel is considered. The kernel can be that of a fractional integral. The integro-differential equation is discretized using the discontinuous Galerkin method with piecewise constant basis functions. Sparse quadrature is introduced for the convolution term to overcome the problem with the growing amount of data that has to be stored and used in each time-step. A priori and a posteriori error estimates are proved. An adaptive strategy based on the a posteriori error estimate is developed. Finally, the precision and effectiveness of the algorithm are demonstrated in the case that the convolution is a fractional integral. This is done by comparing the numerical solutions with analytical solutions.     

A finite-difference procedure to solve weakly singular integro partial differential equation with space-time fractional derivatives

Engineering with Computers - The main aim of the current paper is to propose an efficient numerical technique for solving space-time fractional partial weakly singular integro-differential...     

Numerical Solution of Weakly Singular Linear Volterra Integro-differential Equations

In this paper, we show that a known technique of smoothing can be successfully employed in the numerical solution of weakly singular linear Volterra integro-differential equations and we introduce a Nyström-type method whose order of convergence is also estimated.     

Quasi-wavelet based numerical method for fourth-order partial integro-differential equations with a weakly singular kernel

In this paper, we study the numerical solution of initial boundary-value problem for the fourth-order partial integro-differential equations with a weakly singular kernel. We use the forward Euler scheme for time discretization and the quasi-wavelet based numerical method for space discretization. Detailed discrete formulations are given to the treatment of three different boundary conditions, including clamped-type condition, simply supported-type condition and a transversely supported-type condition. Some numerical experiments are included to demonstrate the validity and applicability of the discrete technique. The comparisons of present results with analytical solutions show that the quasi-wavelet based numerical method has a distinctive local property. Especially, the method is easy to implement and produce very accurate results.     

Finite central difference/finite element approximations for parabolic integro-differential equations

We study the numerical solution of an initial-boundary value problem for parabolic integro-differential equation with a weakly singular kernel. The main purpose of this paper is to construct and analyze stable and high order scheme to efficiently solve the integro-differential equation. The equation is discretized in time by the finite central difference and in space by the finite element method. We prove that the full discretization is unconditionally stable and the numerical solution converges to the exact one with order O(Δt ² + h ^l ). A numerical example demonstrates the theoretical results.     

Fractional convolution quadrature based on generalized Adams methods

In this paper we present a product quadrature rule for Volterra integral equations with weakly singular kernels based on the generalized Adams methods. The formulas represent numerical solvers for fractional differential equations, which inherit the linear stability properties already known for the integer order case. The numerical experiments confirm the valuable properties of this approach.     

An algorithm based on Q.I. modified splines for singular integral models

In this paper, we propose an algorithm supporting an approximation using quasi-interpolatory (q.i.) splines for the numerical solution of integro-differential equations with Cauchy singular kernel. Some different choices of parameters, defining the numerical model, are analyzed and compared in view of the algorithm efficiency.     

A new approach for numerical solution of integro-differential equations via Haar wavelets

A new method is proposed for numerical solution of Fredholm and Volterra integro-differential equations of second kind. The proposed method is based on Haar wavelets approximation. Special characteristics of Haar wavelets approximation has been used in the derivation of this method. The new method is the extension of the recent work [Aziz and Siraj-ul-Islam, New algorithms for numerical solution of nonlinear Fredholm and Volterra integral equations using Haar wavelets, J. Comput. Appl. Math. 239 (2013), pp. 333–345] from integral equations to integro-differential equations. The method is specifically derived for nonlinear problems. Two new algorithms are also proposed based on this new method, one each for numerical solution of Fredholm and Volterra integro-differential equations. The proposed algorithms are generic and are applicable to all types of both nonlinear Fredholm and Volterra integro-differential equations of second kind. The cost of the new algorithms is considerably reduced by using the Broyden's method instead of Newton's method for solution of system of nonlinear equations. Most of the numerical methods designed for solution of integro-differential equations rely on some other technique for numerical integration. The advantage of our method is that it does not use numerical integration. The integrand is approximated using Haar wavelets approximation and then exact integration is performed. The method is tested on number of problems and numerical results are compared with existing methods in the literature. The numerical results indicate that accuracy of the obtained solutions is reasonably high even when the number of collocation points is small.     

An adaptive Huber method with local error control,for the numerical solution of the first kind Abel integral equations

In contrast to the existing plethora of adaptive numerical methods for differential and integro-differential equations, there seems to be a shortage of adaptive methods for purely integral equations with weakly singular kernels, such as the first kind Abel equation. In order to make up this deficiency, an adaptive procedure based on the product-integration method of Huber is developed in this work. In the procedure, an a posteriori estimate of the dominant expansion term of the local discretisation error at a given grid node is used to determine the size of the next integration step, in a way similar to the adaptive solvers for ordinary differential equations. Computational experiments indicate that in practice the control of the local errors is sufficient for bringing the true global errors down to the level of a prescribed error tolerance. The lower limit of the acceptable values of the error tolerance parameter depends on the interference of machine errors, and the quality of the approximations available for the method coefficients specific for a given kernel function.      

Efficient solution of a class of partial integro-differential equation in reproducing kernel space

In [Y.l. Wang, T. Chaolu, Z. Chen, Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems, Int. J. Comput. Math. 87(2) (2010), pp. 367–380], we present three algorithms to solve a class of ordinary differential equations boundary value problems in reproducing kernel space. It is worth noting that our methods can get the solution of partial integro-differential equation. In this note, we use method 2 [M. Dehghan, Solution of a partial integro-differential equation arising from viscoelasticity, Int. J. Comput. Math. 83(1) (2006), pp. 123–129] to solve a class of partial integro-differential equation in reproducing kernel space. Numerical example shows our method is effective and has high accuracy.     

An efficient fractional-order wavelet method for fractional Volterra integro-differential equations

In this paper, an efficient and robust numerical technique is suggested to solve fractional Volterra integro-differential equations (FVIDEs). The proposed method is mainly based on the generalized fractional-order Legendre wavelets (GFLWs), their operational matrices and the Collocation method. The main advantage of the proposed method is that, by using the GFLWs basis, it can provide more efficient and accurate solution for FVIDEs in compare to integer-order wavelet basis. A comparison between the achieved results confirms accuracy and superiority of the proposed GFLWs method for solving FVIDEs. Error analysis and convergence of the GFLWs basis is provided.     

An Implementation of Fast Wavelet Galerkin Methods for Integral Equations of the Second Kind

We present a numerical implementation of the fast Galerkin method for Fredholm integral equations of the second kind using the piecewise polynomial wavelets. We focus on addressing critical issues for the numerical implementation of such a method. They include a choice of practical truncation strategy, numerical integration of weakly singular integrals and the error control of the numerical quadrature. We also implement a multiscale iteration method for solving the resulting compressed linear system. Numerical examples are given to demonstrate the proposed ideas and methods.     

Taylor wavelet method for fractional delay differential equations

We present a new numerical method for solving fractional delay differential equations. The method is based on Taylor wavelets. We establish an exact formula to determine the Riemann–Liouville fractional integral of the Taylor wavelets. The exact formula is then applied to reduce the problem of solving a fractional delay differential equation to the problem of solving a system of algebraic equations. Several numerical examples are presented to show the applicability and the effectiveness of this method.

     

Application of Legendre wavelets for solving fractional differential equations

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.     

An efficient method based on the second kind Chebyshev wavelets for solving variable-order fractional convection diffusion equations

In this paper, a class of variable-order fractional convection diffusion equations have been solved with assistance of the second kind Chebyshev wavelets operational matrix. The operational matrix of variable-order fractional derivative is derived for the second kind Chebyshev wavelets. By implementing the second kind Chebyshev wavelets functions and also the associated operational matrix, the considered equations will be reduced to the corresponding Sylvester equation, which can be solved by some appropriate iterative solvers. Also, the convergence analysis of the proposed numerical method to the exact solutions and error estimation are given. A variety of numerical examples are considered to show the efficiency and accuracy of the presented technique.     

The approximate solution of charged particle motion equations in oscillating magnetic fields using the local multiquadrics collocation method

The charged particle motion for certain configurations of oscillating magnetic fields can be simulated by a Volterra integro-differential equation of the second order with time-periodic coefficients. This paper investigates a simple and accurate scheme for computationally solving these types of integro-differential equations. To start the method, we first reduce the integro-differential equations to equivalent Volterra integral equations of the second kind. Subsequently, the solution of the mentioned Volterra integral equations is estimated by the collocation method based on the local multiquadrics formulated on scattered points. We also expand the proposed method to solve fractional integro-differential equations including non-integer order derivatives. Since the offered method does not need any mesh generations on the solution domain, it can be recognized as a meshless method. To demonstrate the reliability and efficiency of the new technique, several illustrative examples are given. Moreover, the numerical results confirm that the method developed in the current paper in comparison with the method based on the globally supported multiquadrics has much lesser volume computing.

     

A Nyström method for Fredholm integral equations with right-hand sides having isolated singularities

Fredholm integral equations on the interval [?1,1] with right-hand sides having isolated singularities are considered. The original equation is reduced to an equivalent system of Fredholm integral equations with smooth input functions. The Nyström method is applied to the system after a polynomial regularization. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.     

Hybrid Bernstein Block–Pulse functions for solving system of fractional integro-differential equations

This paper deals with the numerical solution of system of fractional integro-differential equations. In this work, we approximate the unknown functions based on the hybrid Bernstein Block–Pulse functions, in conjunction with the collocation method. We introduce the Riemann–Liouville fractional integral operator for the hybrid Bernstein Block–Pulse functions. This operator will be approximated by the Gauss quadrature formula with respect to the Legendre weight function and then it is utilized to reduce the solution of the fractional integro-differential equations to a system of algebraic equations. This system can be easily solved by any usual numerical methods. The existence and uniqueness of the solution have been discussed. Moreover, the convergence analysis of this algorithm will be shown by preparing some theorems. Numerical experiments are presented to show the superiority and efficiency of proposed method in comparison with some other well-known methods.     

An <Emphasis Type="Italic">hp</Emphasis>-version Spectral Collocation Method for Nonlinear Volterra Integro-differential Equation with Weakly Singular Kernels

In this paper, we present an hp-version Legendre–Jacobi spectral collocation method for the nonlinear Volterra integro-differential equations with weakly singular kernels. We derive hp-version error bounds of the collocation method under the \(H^1\)-norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. Numerical experiments demonstrate the effectiveness of the proposed method.