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Mean value theorems for both derivatives and integrals are very useful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. In this article, a semi-analytical method that is based on weighted mean-value theorem for obtaining solutions for a wide class of Fredholm integral equations of the second kind is introduced. Illustrative examples are provided to show the significant advantage of the proposed method over some existing techniques.

In this article, we consider the second-type linear Volterra integral equations whose kernels based upon the difference of the arguments. The aim is to convert the integral equation to an algebraic one. This is achieved by approximating functions appearing in the integral equation with the Bernstein polynomials. Since the kernel is of convolution type, the integral is represented as a convolution product. Taylor expansion of kernel along with the properties of convolution are used to represent the integral in terms of the Bernstein polynomials so that a set of algebraic equations is obtained. This set of algebraic equations is solved and approximate solution is obtained. We also provide a simple algorithm which depends both on the degree of the Bernstein polynomials and that of monomials. Illustrative examples are provided to show the validity and applicability of the method.

Abstract:In this work, we consider two-dimensional linear and nonlinear Fredholm integral equations of the first kind. The combination of the regularization method and the homotopy perturbation method, or shortly, the regularization-homotopy method is used to find a solution to the equation. The application of this method is based upon converting the first kind of equation to the second kind by applying the regularization method. Then the homotopy perturbation method is employed to the resulting second kind of equation to obtain a solution. A few examples including linear and nonlinear equations are provided to show the validity and applicability of this approach.

Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article we show how an algorithm of Madych for scalar wavelets can be generalized to multiwavelets, given an extra assumption. We then develop a new algorithm that does not require this additional condition. Finally, we apply results from a previous paper to resolve the non-uniqueness of the algorithm by imposing regularity conditions (including approximation orders) on the boundary functions