Modified Homotopy Perturbation Method For Solving High-Order Integro-Differential Equation

In this work, a new modification of homotopy perturbation method was proposed to find analytical solution of high-order integro-differential equations. The Modification process yields the Taylor series of the exact solution. Canonical polynomials are used as basis function. The assumed approximate solution was substituted into the problem considered in which the coefficients of the homotopy perturbation parameter p were compared, and then solved, resulting to a single algebraic equation. Thus, algebraic linear system of equations were obtained by equating the coefficients of various powers of the independent variables in the equation to zero,  which are then solved simultaneously using MAPLE 18 software to obtain the values of the unknown constants in the equations. The values of the unknown constants were substituted back to get the initial approximation which yield the final solution. Some examples were given to illustrate the effectiveness of the method. Keywords: Homotopy perturbation, Integro-differential equation, Canonical polynomial, Basis functio

On the Regularization-Homotopy Analysis Method for Linear and Nonlinear Fredholm Integral Equations of the First Kind

Fredholm integral equations of the first kind are considered by applying regularization method and the homotopy analysis method. This kind of integral equations are considered as an ill-posed problem and for this reason needs an effective method in solving them. This method first transforms a given Fredholm integral equation of the first kind to the second kind by the regularization method and then solves the transformed equation using homotopy analysis method. Approximation of the solution will be of much concern since it is not always the case to get the solution to converge and the existence of the solution is not always guaranteed as this kind of Fredholm integral equation is not well-posed

Solution of Fuzzy Fredholm Integral Equation via Modified Homotopy Method

In this paper, we proposed a modification to the Homotopy method by introducing accelerating parameters for solving fuzzy integral equations.The modified method is employed to find exact solutions for fuzzy Fredholm integral equations . The results imply that the modified method is very simple and effective

On FIDEs System by Modified Sumudu Decomposition Method

In this paper, the technique of modified Sumudu decomposition method has been employed to solve a system of Fredholm integro-differential equations with initial conditions. Two examples are discussed to show applicability, reliability and the performance of the modified sumudu decomposition method. This study showed the capability, simplicity and effectiveness of the modified approach. Keywords: Modified Sumudu decomposition method; System of Fredholm integro-differential equations

A hybrid iterative scheme for optimal control problems governed by some Fredholm Integral Equations

This paper presents an iterative approach based on hybrid of perturbation and parametrization methods for obtaining approximate solutions of optimal control problems governed by some Fredholm integral equations. By somenumerical examples, it is emphasized that this scheme is very effective and it produces approximate solutions with high precision. Convergence of the given iterative scheme is also discussed

Application of Exponentially Fitted Collocation Algorithm for Solving nth-Order Fredholm Integrodifferential Equations

In this paper, we seek to build and apply an exponentially fitted collocation algorithm (EFCA) for the solutions of nth-order Fredholm type integrodifferential equations. For this purpose, an EFCA was formulated and applied to solve four examples from the literature. Numerical experiment was performed and the results were compared with the exact solutions, and some existing methods. From the four examples considered, the results obtained showed that the proposed algorithm is fast, efficient, and reliable.

Solution Properties for Pertubed Linear and Nonlinear Integrals Equations

In this study we consider perturbative series solution with respect to a parameter {\epsilon} > 0. In this methodology the solution is considered as an infinite sum of a series of functional terms which usually converges fast to the exact desired solution. Then we investigate perturbative solutions for kernel perturbed integral equations and prove the convergence in an appropriate ranges of the perturbation series. Next we investigate perturbation series solutions for nonlinear perturbations of integral equations of Hammerstein type and formulate conditions for their convergence. Finally we prove the existence of a maximal perturbation range for non linear integral equations