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

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

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

Approximate Solution of Nonlinear fractional Integro-Di erential Equations By He's Homotopy Perturbation Method And The Modi cation of He's Variational Iteration Method

In this paper, we compare the modi cation of He's variational iteration method (MVIM), and He's homotopy perturbation method (HPM), in order to obtain the approximate solution of nonlinear frac- tional integro-di erential equations of Volterra and Fredholm integro-di erential equations, we present some examples to nd out accuracy of the methods. keywords: Fractional integro-di erential equations, Caputo derivative, modi cation of He's variational iteration method, homotopy perturbation metho

Study on Solving Two-dimensional Linear and Nonlinear Volterra Partial Integro-differential Equations by Reduced Differential Transform Method

In this article, we study on the analytical and numerical solution of two-dimensional linear and nonlinear Volterra partial integro-differential equations with the appropriate initial condition by means of reduced differential transform method. The advantage of this method is its simplicity in using, it solves the problem directly without the need for linearization, perturbation, or any other transformation and gives the solution in the form of convergent power series with elegantly computed components. The validity and efficiency of this method are illustrated by considering five computational examples

Numerical Algorithm for Nonlinear Delayed Differential Systems of nnth Order

The purpose of this paper is to propose a semi-analytical technique convenient for numerical approximation of solutions of the initial value problem for $p$-dimensional delayed and neutral differential systems with constant, proportional and time varying delays. The algorithm is based on combination of the method of steps and the differential transformation. Convergence analysis of the presented method is given as well. Applicability of the presented approach is demonstrated in two examples: A system of pantograph type differential equations and a system of neutral functional differential equations with all three types of delays considered. Accuracy of the results is compared to results obtained by the Laplace decomposition algorithm, the residual power series method and Matlab package DDENSD. Comparison of computing time is done too, showing reliability and efficiency of the proposed technique.Comment: arXiv admin note: text overlap with arXiv:1501.00411 Author's reply: the text overlap may be caused by the fact that this article is concerning systems of equations, while the other paper was about single equation

Numerical computational approach for 6th order boundary value problems

This study introduces numerical computational methods that employ fourth-kind Chebyshev polynomials as basis functions to solve sixth-order boundary value problems. The approach transforms the BVPs into a system of linear algebraic equations, expressed as unknown Chebyshev coefficients, which are subsequently solved through matrix inversion. Numerical experiments were conducted to validate the accuracy and efficiency of the technique, demonstrating its simplicity and superiority over existing solutions. The graphical representation of the method's solution is also presented

(SI10-123) Comparison Between the Homotopy Perturbation Method and Variational Iteration Method for Fuzzy Differential Equations

In this article, the authors discusses the numerical simulations of higher-order differential equations under a fuzzy environment by using Homotopy Perturbation Method and Variational Iteration Method. The fuzzy parameter and variables are represented by triangular fuzzy convex normalized sets. Comparison of the results are obtained by the homotopy perturbation method with those obtained by the variational iteration method. Examples are provided to demonstrate the theory