Unique solvability of boundary value problem for functional differential equations with involution

In this paper we consider a boundary value problem for systems of Fredholm type integral-differential equations with involutive transformation, containing derivative of the required function on the right-hand side under the integral sign. Applying properties of an involutive transformation, original boundary value problem is reduced to a boundary value problem for systems of integral-differential equations, containing derivative of the required function on the right side under the integral sign. Assuming existence of resolvent of the integral equation with respect to the kernel K˜2(t, s) (this is the kernel of the integral equation that contains the derivative of the desired function) and using properties of the resolvent, integral-differential equation with a derivative on the right-hand side is reduced to a Fredholm type integral-differential equation, in which there is no derivative of the desired function on the right side of the equation. Further, the obtained boundary value problem is solved by the parametrization method created by Professor D. Dzhumabaev. Based on this method, the problem is reduced to solving a special Cauchy problem with respect to the introduced new functions and to solving systems of linear algebraic equations with respect to the introduced parameters. An algorithm to find a solution is proposed. As is known, in contrast to the Cauchy problem for ordinary differential equations, the special Cauchy problem for systems of integral-differential equations is not always solvable. Necessary conditions for unique solvability of the special Cauchy problem were established. By using results obtained by Professor D. Dzhumabaev, necessary and sufficient conditions for the unique solvability of the original problem were established.

Approximation of Fixed Points of Multivalued Generalized (α,β)-Nonexpansive Mappings in an Ordered CAT(0) Space

The purpose of this article is to initiate the notion of monotone multivalued generalized (α,β)-nonexpansive mappings and explore the iterative approximation of the fixed points for the mapping in an ordered CAT(0) space. In particular, we employ the S-iteration algorithm in CAT(0) space to prove some convergence results. Moreover, some examples and useful results related to the proposed mapping are provided. Numerical experiments are also provided to illustrate and compare the convergence of the iteration scheme. Finally, an application of the iterative scheme has been presented in finding the solutions of integral differential equation.

Analytical Methods for Nonlinear Evolution Equations in Mathematical Physics

In this article, we will apply some of the algebraic methods to find great moving solutions to some nonlinear physical and engineering questions, such as a nonlinear (1 + 1) Ito integral differential equation and (1 + 1) nonlinear Schrödinger equation. To analyze practical solutions to these problems, we essentially use the generalized expansion approach. After various W and G options, we get several clear means of estimating the plentiful nonlinear physics solutions. We present a process like-direct expansion process-method of expansion. In the particular case of W′=λG, G′=μW in which λ and μ are arbitrary constants, we use the expansion process to build some new exact solutions for nonlinear equations of growth if it fulfills the decoupled differential equations.

On the Connection of Cascade-Probability Function on the Formation of Primary-Knocked on Atoms for Ions Taking into Account Energy Losses with a Boltzman Equation

The integral-differential equation of the cascade process for ions was solved using the Laplace transform and the method of successive approximations, taking into account the energy loss during the formation of primary-knocked-on atoms (PKA) in a one-dimensional model of an elementary atom. It is shown that the solution includes a cascade-probability function (CPF) for these particles. The main properties of CPF are considered and its graphical dependencies on the depth of registration are presented. It is shown that with the specific ionization loss coefficient k = 0, the FQM turns into the simplest cascade-probability function. When λ0→ 0, λ0→∞ and n→∞, the KV-function is equal to 0. The sum of the probabilities for all possible collisions from 0 to ∞ is 1. As the detection depth h increases, for all values of n, the CRF increases, reaches a maximum and then decreases . With increasing n, the curves shift to the right.

Difference schemes for the semilinear integral-differential equation of the hyperbolic type

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The first order and the second order of accuracy difference schemes approximately solving this problem are presented. The convergence estimates for the solutions of these difference schemes are obtained. Theoretical results are supported by numerical example.