in this paper, the two-dimensional triangular orthogonal functions (2d-tfs) are applied for solving a class of nonlinear two-dimensional volterra integral equations. 2d-tfs method transforms these integral equations into a system of linear algebraic equations. the high accuracy of this method is verified through a numerical example and comparison of the results with the other numerical methods.

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...

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...

The nonlinear integral equations arise in the theory of parabolic boundary value problems, engineering, various mathematical physics, and theory of elasticity 1–3 . In recent years, several analytical and numerical methods of this kind of problems have been presented 4, 5 . Analytically, the decomposition methods are used in 6, 7 . The classical method of successive approximations was introduce...

In this paper, to solve a linear one-dimensional Volterra  integral equation of the second kind. For this purpose using the equation form, we have defined a linear transformation and by using it's conjugate and reproducing kernel functions, we obtain a basis for the functions space.Then we obtain the solution of  integral equation in terms of the basis functions. The examples presented in this ...

Degenerate kernel approximation method is generalized to solve Hammerstein system of Fredholm integral equations of the second kind. This method approximates the system of integral equations by constructing degenerate kernel approximations and then the problem is reduced to the solution of a system of algebraic equations. Convergence analysis is investigated and on some test problems, the propo...

Although perturbation techniques are widely applied to analyze nonlinear problems in science and engineering, they are however so strongly dependent on small parameters appeared in equations under consideration that they are restricted only to weakly nonlinear problems. For strongly nonlinear problems which don’t contain any small parameters, perturbation techniques are invalid. So, it seems ne...