Sumudu Decomposition Method for Solving Fuzzy Integro-Differential Equations

Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations

In this paper, fuzzy neural network (FNN) can be trained with crisp and fuzzy data. The work of this paper is an expansion of the research of fuzzy linear Fredholm integro-differential equations. In this work, we interpret a fuzzy integro-differential equations. In this paper, a novel hybrid method based on FNN and Newton–Cotes methods with positive coefficient for the solution of system of fuzzy linear Fredholm integro-differential equations of the second kind with fuzzy initial values is presented. The problem formulation of the proposed UA is quite straightforward. To obtain the “best-approximated” solution of system of fuzzy linear Fredholm integro-differential equations, the adjustable parameters of PFNN are systematically adjusted by using the learning algorithm. Within this paper, the fuzzy neural network model is used to obtain an estimate for the fuzzy parameters in a statistical sense. Based on the extension principle, a simple algorithm from the cost function of the fuzzy neural network is proposed, in order to find the approximate parameters. We propose a learning algorithm from the cost function for adjusting of fuzzy weights. Here neural network is considered as a part of a larger field called neural computing or soft computing. Finally, we illustrate our approach by some numerical examples.

In this paper, we interpret a nonlinear fuzzy Fredholm integro-differential equations by using the strongly generalized differentiability concept. Based on the parametric form of a fuzzy number, a fuzzy integro-differential equation converts to two systems of integro-differential equations in the crisp case. Also, we use the parametric form of fuzzy number, and an iterative approach for obtaining approximate solution for a class of nonlinear fuzzy Fredholm integro-differential equation of the second kind is proposed. This paper presents a method based on Newton–Cotes methods with positive coefficient. Then we obtain approximate solution of the nonlinear fuzzy integro-differential equations by an iterative approach.

The idea of this research is to find approximate solution of Linear Volterra Integro-Differential Equation (LVIDE) of the first order and second kind by using different degrees of the Touchard polynomials is presented. Because the simplicity definition applying this method will give high resolution results in low cost and short time, also can be easily differentiated, integrated and converge to an exact solution.The algorithm and example given are to illustrate the solution in this way and compare it with the exact solution.

In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.

In this paper, by using the collocation method the operational matrix of Caputo fractional derivative was constructed for the cubic B-spline scaling function and wavelets. We apply the operational matrix and the Collocation method to solve linear fractional Volterra integro-differential equations. By using the principal characteristic of this technique, we have the operational matrix, which is applied to solve any linear fractional Volterra integro-differential equation by reducing the time, computer memory occupation and convert to a system of linear equations. Finally, the validity and applicability of the new technique will be shown by some numerical examples and convergence analysis.

Fractional delay integrodifferential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

In this paper, we use the variational iteration method (VIM) to construct approximate solutions for the general [Formula: see text]th-order integro-differential equations. We show that his method can be effectively and easily used to solve some classes of linear and nonlinear Volterra integro-differential equations. Finally, some numerical examples with exact solutions are given.

Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients

This paper is dealing with non-polynomial spline functions "generalized spline" to find the approximate solution of linear Volterra integro-differential equations of the second kind and extension of this work to solve system of linear Volterra integro-differential equations. The performance of generalized spline functions are illustrated in test examples

A numerical solution is presented for the solution of integrodifferential equations involving convolution integrals. These equations can be viewed as a generalization of the fractional differential equations, where the convolution integral represents the fractional derivative. The presented method is based on the concept of the analog equation, which converts the integrodifferential equation into a single-term ordinary differential equation, the analog equation. The latter is solved using an integral equation method. The method applies to linear and nonlinear integrodifferential equations with constant and variable coefficients as well. The resulting numerical scheme is stable, second-order accurate and simple to program on a computer. Several numerical examples with different kernels are presented, which demonstrate the efficiency and accuracy of the method showing thus the excellent performance of the numerical scheme.

In this paper we study the controllability for the nonlinear fuzzy integro-differential equations on <TEX>$E_N^n$</TEX> by using the concept of fuzzy number of dimension n whose values are normal, convex, upper semicontinuous and compactly supported surface in <TEX>$R^n$</TEX>. <TEX>$E_N^n$</TEX> the set of all fuzzy numbers in <TEX>$R^n$</TEX> with edges having bases parallel to axis <TEX>$X_1,\;X_2, ... , X_n$</TEX>.

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (J. Comput. Appl. Math. 290:633–640, 2015).

Journal Article A posteriori error estimates of discontinuous Galerkin methods for non-standard Volterra integro-differential equations Get access Jingtang Ma, Jingtang Ma **Email: jingtang@lsec.cc.ac.cn Search for other works by this author on: Oxford Academic Google Scholar Hermann Brunner Hermann Brunner Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 26, Issue 1, January 2006, Pages 78–95, https://doi.org/10.1093/imanum/dri024 Published: 01 January 2006

ABSTRACTThe main purpose of this work is to provide a new direct numerical method for high-order linear Volterra integro-differential equations (VIDEs). An algorithm based on the use of Taylor polynomials is developed for the numerical solution of high-order linear VIDEs. It is shown that this algorithm is convergent. Numerical results are presented and comparisons are made with well-known numerical methods to prove the effectiveness of the presented algorithm.