An Extended Numerical Method by Stancu Polynomials for Solution of Integro-Differential Equations Arising in Oscillating Magnetic Fields

In this study, the Bernstein collocation method has been expanded to Stancu collocation method for numerical solution of the charged particle motion for certain configurations of oscillating magnetic fields modelled by a class of linear integro-differential equations. As the method has been improved, the Stancu polynomials that are generalization of the Bernstein polynomials have been used. The method has been tested on a physical problem how the method can be applied. Moreover, numerical results of the method have been compared with the numerical results of the other methods to indicate the efficiency of the method.

THE WELL-POSEDNESS OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS IN COMPLEX BANACH SPACES

Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).

Galerkin Method for Numerical Solution of Volterra Integro-Differential Equations with Certain Orthogonal Basis Function

This paper concerns the implementation of the orthogonal polynomials using the Galerkin method for solving Volterra integro-differential and Fredholm integro-differential equations. The constructed orthogonal polynomials are used as basis functions in the assumed solution employed. Numerical examples for some selected problems are provided and the results obtained show that the Galerkin method with orthogonal polynomials as basis functions performed creditably well in terms of absolute errors obtained.

Application of HAM for Nonlinear Integro-Differential Equations of Order Two

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable;therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type;then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

Highly efficient H^1-Galerkin mixed finite element method (MFEM) for parabolic integro-differential equation

A highly efficient H1-Galerkin mixed finite element method （MFEM） is presented with linear triangular element for the parabolic integro-differential equation. Firstly, some new results about the integral estimation and asymptotic expansions are studied. Then, the superconvergence of order O（h^2） for both the original variable u in H1 （Ω） norm and the flux p = u in H（div, Ω） norm is derived through the interpolation post processing technique. Furthermore, with the help of the asymptotic expansions and a suitable auxiliary problem, the extrapolation solutions with accuracy O（h^3） are obtained for the above two variables. Finally, some numerical results are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method.

CONVERGENCE ANALYSIS OF THE JACOBI SPECTRAL-COLLOCATION METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L^∞ norm and weighted L^2-norm. The numerical examples are given to illustrate the theoretical results.

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.

MULTIPLE POSITIVE SOLUTIONS FOR FIRST ORDER IMPULSIVE SINGULAR INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF LINE

In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

Solving Fractional Integro-Differential Equations by Using Sumudu Transform Method and Hermite Spectral Collocation Method

In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.

ON MONOTONE ITERATIVE METHOD FOR INITIAL VALUE PROBLEMS OF NONLINEAR SECOND-ORDER INTEGRO-DIFFERENTIAL EQUATIONS IN BANACH SPACES

Using the monotone iterative method and Monch Fixed point theorem, the existence of solutions and coupled minimal and maximal quasisolutions of initial value problems for mixed monotone second-order integro-differential equations in Banach spaces are studied. Some existence theorems of solutions and coupled minimal and maximal quasisolutions are obtained.

MULTIPLE POSITIVE SOLUTIONS FOR FIRST ORDER IMPULSIVE SUPERLINEAR INTEGRO-DIFFERENTIAL EQUATIONS ON THE HALF LINE

In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive superlinear integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

EXISTENCE AND UNIQUENESS RESULTS FOR BOUNDARY VALUE PROBLEMS OF HIGHER ORDER FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS INVOLVING GRONWALL'S INEQUALITY IN BANACH SPACES

We study boundary value problems for fractional integro-differential equations involving Caputo derivative of order α∈ （n-1, n） in Banach spaces. Existence and uniqueness results of solutions are established by virtue of the Holder＇s inequality, a suitable singular Cronwall＇s inequality and fixed point theorem via a priori estimate method. At last, examples are given to illustrate the results.

DISCRETE GALERKIN METHOD FOR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

In this article, we develop a fully Discrete Galerkin（DG） method for solving ini- tial value fractional integro-differential equations（FIDEs）. We consider Generalized Jacobi polynomials（CJPs） with indexes corresponding to the number of homogeneous initial conditions as natural basis functions for the approximate solution. The fractional derivatives are used in the Caputo sense. The numerical solvability of algebraic system obtained from implementation of proposed method for a special case of FIDEs is investigated. We also provide a suitable convergence analysis to approximate solutions under a more general regularity assumption on the exact solution. Numerical results are presented to demonstrate the effectiveness of the proposed method.

A Differential Quadrature Based Approach for Volterra Partial Integro-Differential Equation with a Weakly Singular Kernel

Differential quadrature method is employed by numerous researchers due to its numerical accuracy and computational efficiency,and is mentioned as potential alternative of conventional numerical methods.In this paper,a differential quadrature based numerical scheme is developed for solving volterra partial integro-differential equation of second order having a weakly singular kernel.The scheme uses cubic trigonometric B-spline functions to determine the weighting coefficients in the differential quadrature approximation of the second order spatial derivative.The advantage of this approximation is that it reduces the problem to a first order time dependent integro-differential equation(IDE).The proposed scheme is obtained in the form of an algebraic system by reducing the time dependent IDE through unconditionally stable Euler backward method as time integrator.The scheme is validated using a homogeneous and two nonhomogeneous test problems.Conditioning of the system matrix and numerical convergence of the method are analyzed for spatial and temporal domain discretization parameters.Comparison of results of the present approach with Sinc collocation method and quasi-wavelet method are also made.

Superconvergence of Finite Element Approximations to Parabolic and Hyperbolic Integro-Differential Equations

The object of this paper is to investigate the superconvergence properties of finite element approximations to parabolic and hyperbolic integro-differential equations. The quasi projection technique introduced earlier by Douglas et al. is developed to derive the O(h2r) order knot superconvergence in the case of a single space variable, and to show the optimal order negative norm estimates in the case of several space variables.

A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations

The aim of this paper is to obtain the numerical solutions of fractional Volterra integrodifferential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points.We convert the fractional order integro-differential equation into integral equation by fractional order integral,and transfer the integro equations into a system of linear equations by the Gausssian quadrature.We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in L∞norm.Two numerical examples demonstrate the high accuracy and fast convergence of the method at last.

Boundary Value Problems for Second-order Integro-differential Equations in Banach Spaces

This paper investigates the maximal and minimal solutions of initial value problems for second order nonlinear integro-differential equations of Volterra type on a finite interval in a Banach space by establishing a comparison result and using the monotone iterative technique.

A SPECTRAL METHOD FOR A WEAKLY SINGULAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATION WITH PANTOGRAPH DELAY

In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation(PIDE)with a weakly singular kernel.Cubic trigonometric B-spline(CTBS)functions are used for interpolation in both methods.The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations.The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values.An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method.An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method.The methods are tested with three nonhomogeneous problems for their validation.Stability,computational efficiency and numerical convergence of the methods are analyzed.Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made.Convection and diffusion dominant cases are discussed in terms of Peclet number.The results are also compared with cubic B-spline collocation method.

PERIODIC BOUNDARY VALUE PROBLEMS FOR FIRST-ORDER INTEGRO-DIFFERENTIAL EQUATIONS OF MIXED TYPE

The existence of at feast one solution and the existence of extreme solutions of periodic boundary value problems for first-order integro-differential equations of mixed type are studied, in the presence of generalized upper and laver solutions. The discussion is based an new comparison theorems and coincidence degree and monotone iterative methods.