Approximate solutions of Fredholm integral equations of the second kind

This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail.     

On the existence of asymptotically stable solutions of certain integral equations

In this paper two existence results of asymptotically stable solutions of certain integral equations are presented.     

Existence of solutions of certain nonconvex optimal control problems governed by nonlinear integral equations

Optimal control problems with nonlinear equations usually do not have a solution, i.e. an optimal control. Nevertheless, if the cost functional is uniformly concave with respect to the state, the solution may exist. Using the Balder's technique based on a Youngmeasure relaxation, Bauer's external principle and investigation of extreme Young measures; the existence is demonstrated here for optimal control processes described by nonlinear integral equations     

Numerical solutions for functional integral equations

Convergence criteria are given for a family of numerical methodsfor functional integral equations with delay. Several numericalmethods are compared.     

Approximate solution for a class of hypersingular integral equations

A simple approximate method for solving a general hypersingular integral equation of the first kind with its kernel consisting of a hypersingular part and a regular part is developed here. The method is illustrated by considering some simple examples.     

Approximate solutions of abstract differential equations

The methods of arbitrarily high orders of accuracy for the solution of an abstract ordinary differential equation are studied. The right-hand side of the differential equation under investigation contains an unbounded operator which is an infinitesimal generator of a strongly continuous semigroup of operators. Necessary and sufficient conditions are found for a rational function to approximate the given semigroup with high accuracy. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Approximate solutions of nonlinear fractional equations

Fractional exponential that are invariant under fractional derivatives, elementary and special fractional functions are introduced. Approximate solutions to fractional Burgers equation, by using the homotopy perturbation method, are obtained. Furthermore, real integral representations for some H-functions are found that may be very helpful in numerical computations.     

Approximate solutions of functional differential equations

The author discusses the best approximate solution of the functional differential equation x′(t) = F(t, x(t), x(h(t))), 0 < t < l satisfying the initial condition x(0) = x₀, where x(t) is an n-dimensional real vector. He shows that, under certain conditions, the above initial value problem has a unique solution y(t) and a unique best approximate solution $\text{p}\text{?}{}_{\mathrm{k}}\text{(t)}$ of degree k (cf. [1]) for a given positive integer k. Furthermore, $\text{sup}{}_{\mathrm{0?t?l}}\text{¦}\text{p}\text{?}{}_{\mathrm{k}}\text{(t) ? y(t)¦ → 0}\text{as}\text{k → ∞}$, where ¦ · ¦ is any norm in Rⁿ.     

Approximate analytical solutions for Kolmogorov’s equations

This paper reports the explicit analytical solutions for Kolmogorov’s equations. Kolmogorov’s equations are commonly used to describe the structure of local isotropic turbulence, but their exact analytical solutions have not yet been found. In this paper, the closed-form solutions for two kinds of Kolmogorov’s equations are obtained. The derivations of the approximate solutions are based on the homotopy analysis method, which is a new tool for obtaining the approximate analytical solutions of both strong and weak nonlinear differential equations. To examine the validity of the approximate solutions, numerical comparisons between results from the homotopy analysis method and the fourth-order Runge-Kutta method are carried out. It is shown that the results are in good agreement.