Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce

Multiple Perturbed Collocation Tau Method for Solving Nonlinear Integro-Differential Equations

The purpose of the study was to investigate the numerical solution of non-linear Fredholm and Volterra integro-differential equations by the proposed method called Multiple Perturbed Collocation Tau Method (MPCTM). We assumed a perturbed approximate solution in terms of Chebyshev  polynomial basis function and then determined the derivatives of the perturbed approximate solution which are then substituted into the special classes of the problems considered. Thus, resulting into n-folds integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Newton’s method which are then substituted back into the approximate solution.Illustrative examples are given to demonstrate the efficiency, computational cost and accuracy of the method. The results obtained with some numerical examples are compared favorable with some existing numerical methods in literature and with the exact solutions where they are known in closed form.Keywords: Nonlinear Problems, Tau Method, Integro-Differential, Newton’s method

An exponentially convergent Volterra-Fredholm method for integro-differential equations

Extending the authors’ recent work [15] on the explicit computation of error bounds for Nystrom solvers applied to one-dimensional Fredholm integro-differential equations (FIDEs), presented herein is a study of the errors incurred by first transforming (as in, e.g., [21]) the FIDE into a hybrid Volterra-Fredholm integral equation (VFIE). The VFIE is solved via a novel approach that utilises N-node Gauss-Legendre interpolation and quadrature for its Volterra and Fredholm components respectively: this results in numerical solutions whose error converges to zero exponentially with N, the rate of convergence being confirmed via large- N asymptotics. Not only is the exponential rate inherently far superior to the algebraic rate achieved in [21], but also it is demonstrated, via diverse test problems, to improve dramatically on even the exponential rate achieved in [15] via direct Nystrom discretisation of the original FIDE; this improvement is confirmed theoretically

Multistage Homotopy Analysis Method for Solving Nonlinear Integral Equations

In this paper, we present an efficient modification of the homotopy analysis method (HAM) that will facilitate the calculations. We then conduct a comparative study between the new modification and the homotopy analysis method. This modification of the homotopy analysis method is applied to nonlinear integral equations and mixed Volterra-Fredholm integral equations, which yields a series solution with accelerated convergence. Numerical illustrations are investigated to show the features of the technique. The modified method accelerates the rapid convergence of the series solution and reduces the size of work

NUMERICAL SOLUTION OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY AKBARI-GANJI’S METHOD

In this study, Akbari-Ganji’s Method (AGM) was applied to solve Volterra Integro-Differential Difference Equations (VIDDE) using Legendre polynomials as basis functions. Here, a trial solution function of unknown constants that conform with the differential equations together with the initial conditions were assumed and substituted into the equations under consideration. The unknown coefficients are solved for using the new proposed approach, AGM which principally involves the application of the boundary conditions on successive derivatives and integrals of the problem to obtain a system of equations. The system of equation is solved using any appropriate computer software, Maple 18. Some examples were solved and the results compared to the exact solutions

Variational Iteration Method for Mixed Type Integro-Differential Equations

In this paper, we approximate Lagrange multipliers to solve integro differential equations of mixed type those are linear first and second order. It is observed that use of approximate Lagrange multipliers reduces the iteration and give faster results as compare to other techniques. Numerical examples support this idea. Keywords Approximate Lagrange Multiplier, Variational Iteration Method, Mixed Type Integro-Differential Equations