In this paper, we apply the homotopy perturbation method (HPM), modified method(MHPM), variational iteration (VIM), Adomian decomposition (ADM), and (MADM) to solve nonlinear mixed Volterra-Fredholm integral equations itssystem. We investigate approximate solution of equation its system via proposed methods. Thevalidity efficiency these methods are demonstrated through various numerical example...

The mixed Volterra-Fredholm integral equations (VFIEs) arise in various physical and biological models. The main purpose of this article is to propose and analyze efficient Bernoulli collocation techniques for numerically solving classes of two-dimensional linear and nonlinear mixed VFIEs. The novel aspect of the technique is that it reduces the problem under consideration to a system of algebr...

We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schrödinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solutio...

In this paper we tackle on mixed Volterra–Fredholm integral equations, as in linear as in non linear cases. To the aim to obtain numerical solutions of these models, the authors propose in the linear case the direct collocation method using a p-order quasi interpolating spline class and in the nonlinear case the fixed point method based on polynomial approximation built by Schauder tensor bases...

We consider Volterra and Fredholm integral equations for functions with values in L-spaces. This includes corresponding problems for set-valued functions, fuzzy-valued functions and many others. We prove theorems of existence and uniqueness of the solution for such equations and suggest some algorithms for finding approximate solutions. We get initial results in the approximation of functions w...

In this paper, we propose a new modified recursion scheme for the approximate solution of higher-order, multipoint, nonlinear boundary value problems with higher-order Robin-type boundary conditions by the Adomian decomposition method. Our new approach utilizes all of the boundary conditions to derive an equivalent nonlinear Fredholm-Volterra integral equation before establishing the new modifi...

This paper presents a numerical method for solving the stochastic nonlinear volterra-fredholm integral equation (SNVFIE) driven by a standard Brownian motion (SBM). The method is illustrated via a stochastic operational matrix (SOM) based on the triangular functions (TFs) in combination with the collocation method. With using this approach, the SNVFIE reduces to a stochastic nonlinear system of...

In this paper, we compute piecewise constant bounds on the solution of mixed nonlinear Volterra-Fredholm integral equations. The enclosures are in the form of intervals which are guaranteed to contain the exact solution considering all round-off and truncation errors, so the width of interval solutions allows us to control the error estimation. An iterative algorithm to improve the accuracy of ...

This paper discusses the eigenvalue problem of second-order Sturm-Liouville equation. We transform the governing differential equation to the Fredholm-Volterra integral equation with appropriate end supports. By expanding the unknown function into the shifted Chebyshev polynomials, we directly get the corresponding polynomial characteristic equations, where the lower and higher-order eigenvalue...

In this paper, an approach based on statistical spline model (SSM) and collocation method is proposed to solve Volterra-Fredholm integral equations. The set of collocation nodes is chosen so that the points yield minimal error in the nodal polynomials. Under some standard assumptions, we establish the convergence property of this approach. Numerical results on some problems are given to describ...