Solving high-order nonlinear Volterra-Fredholm integro-differential equations by differential transform method

Abstract

In this paper, we apply the differential transformation method to high-order nonlinear Volterra- Fredholm integro-differential equations with se- parable kernels. Some different examples are considered the results of these examples indi-cated that the procedure of the differential transformation method is simple and effective, and could provide an accurate approximate solution or exact solution

Numerical Solution of Nonlinear Fredholm Integrodifferential Equations by Hybrid of Block-Pulse Functions and Normalized Bernstein Polynomials

“…Finding numerical solutions for Fredholm integrodifferential equations is one of the oldest problems in applied mathematics. Numerous works have been focusing on the development of more advanced and efficient methods for solving integrodifferential equations such as wavelets method [4,5], Walsh functions method [6], sinc-collocation method [7], homotopy analysis method [8], differential transform method [9], the hybrid Legendre polynomials and blockpulse functions [10], Chebyshev polynomials method [11], and Bernoulli matrix method [12].…”

Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

“…Recently, many authors have investigated the numerical methods for integral equations. These methods include a cubic spline approximation in C 2 to the solution of the Volterra integral equation of the second kind [33], quintic B-spline method [30], Bernstein operational matrix of derivative [4], hybrid of block pulse functions and normalized Bernstein polynomials [5], iterative method [49], sinc-collocation method [48], bivariate splines on nonuniform partitions [36], Jacobi operational matrices for solving delay or advanced integro-differential equations [40], the tau approximation for the Volterra-Hammerstein integral equations [21], b-spline collocation and cubature formulas [12] and [37], wavelet method [6], Walsh function method [35], Chebyshev finite difference method [13], differential transform method [7], Legendre polynomial method [39], an approximating solution, based on Lagrange interpolation and spline functions, to treat functional integral equations of Fredholm type and Volterra type [20], CAS wavelets method [22], an efficient matrix method based on Bell polynomials for solving nonlinear Fredholm-Volterra integral equations [32], collocation methods [10], Taylor polynomial methods [46], and Bernoulli matrix method [9]. Xuhao Li and Patricia J.Y.…”

Boole Collocation Method Based on Residual Correction for Solving Linear Fredholm Integro-Differential Equation

“…Therefore, the solution of the integro differential equations in the applied mathematics is important. Since the analytical solution of integro differential equations are difficult, numerical methods such as Bernoulli polynomial approach method [1], Adomian's decomposition method [2], a multiscale Galerkin method [3], Chebyshev wavelet method [4], Bessel matrix method [5], the differential transform method [6], Laguerre polynomial solution method [7], Modified decomposition method [8], an exponential method [9], Tau numerical solution method [10], Legendre polynomial solutions method [11], reproducing kernel method [12], hybrid collocation method [13], a moving mesh method [14], Chebyshev pseudo-spectral method [15] are developed [16][17][18][19][20][21][22][23][24][25][26][27]. In this study, Boole method is developed for approximate solution of the mth order linear Fredholm integro differential equation.…”