Picard Boundary Value Problems for Second Order Nonlinear Functional Integro-Differential Equations

Periodic boundary value problems for impulsive integro-differential equations of mixed type

In this work, we consider the second order nonlinear integro-differential Equation (IDEs) of the Volterra-Fredholm type. One of the popular methods for solving Volterra or Fredholm type IDEs is the method of quadrature while the problem of consideration is a linear problem. If IDEs are nonlinear or integral kernel is complicated, then quadrature rule is not most suitable; therefore, other types of methods are needed to develop. One of the suitable and effective method is homotopy analysis method (HAM) developed by Liao in 1992. To apply HAM, we firstly reduced the IDEs into nonlinear integral Equation (IEs) of Volterra-Fredholm type; then the standard HAM was applied. Gauss-Legendre quadrature formula was used for kernel integrations. Obtained system of algebraic equations was solved numerically. Moreover, numerical examples demonstrate the high accuracy of the proposed method. Comparisons with other methods are also provided. The results show that the proposed method is simple, effective and dominated other methods.

In this paper, we investigate the existence of solutions to the boundary value problems for the ordinary integro-differential equations with three-point boundary conditions. Existence and uniqueness results are obtained by using the well-known fixed point theorems.

In this paper, we prove the existence of solutions of initial value problems for nth order nonlinear impulsive integro-differential equations of mixed type on an infinite interval with an infinite number of impulsive times in Banach spaces. Our results are obtained by introducing a suitable measure of noncompactness.

Local and global existence of solutions for a class of integro-differential equations with delay

On the unsteady forces during the motion of a sedimenting particle

In this article, we consider a nonlinear Sobolev type fractional functional integrodifferential equations in a Banach space along with a nonlocal condition. Sufficient conditions for existence, uniqueness and dependence on initial data of local solutions of considered problem are derived by employing fixed point techniques and theory of classical semigroup. Further, we also render the criteria for existence of global solution. At the end, we provide an application to elaborate the obtained results.

In this work, I will discuss the solution of the new in a nonlinear integro-differential equation (NI-DE) form. The NI-DE is established from the nuclear transport equation, which is a linearized derivative of the equation developed by Boltzmann for the kinetic theory of gases. In this study, I will introduce the basic equations and boundary conditions for the formulation of the problem. Then, the NI-DE is established from the planar geometry problem of the neutron transport equation. In addition, I will prove the existence of a unique solution to the problem. In addition, I will use the numerical method to obtain a system of NI-DEs. Therefore, I will prove, under certain conditions, the existence of a unique solution to this system. Finally, I will present two different effective methods to solve the problem numerically, and I will discuss the results.

This study discussed the existence of solutions for a nonlinear fractional integro-differential equation of order with four-point nonlocal integral boundary conditions. The given problem is transformed to an equivalent fixed point problem in terms of an operator equation. Then, by means of Banach contraction principle and a fixed point theorem due to Krasnoselskii, the existence results are obtained. The last existence result is based on nonlinear alternative of Leray-Schauder type. An illustrative example is also presented. Key words: Integro-differential equations, fractional order, four-point integral boundary conditions, fixed point theorems, Leray-Schauder degree.

Solution of boundary value problems for integro-differential equations by using differential transform method

The paper is focused on solvability of an inverse boundary problem with an unknown coefficient which depends on time for a second-order parabolic equation with non-classical boundary conditions. The idea of the problem is that together with the solution it is required to determine the unknown coefficient. The problem is considered in the rectangular area. The paper introduces a classical solution of the set problem. At first, an auxiliary inverse boundary problem is examined and the equivalence (in some sense) of the original problem is proved. First, we apply a method of variable separation to analyze the auxiliary inverse boundary problem. Then, we examine a spectral problem for an ordinary second-order differential equation with integral conditions. Having used a formal scheme of the method of variable separation, the solution of a direct boundary problem (in case of specified unknown function) resolves itself into solution of Cauchy problem. After that the solution is limited to the solution of a countable system of integro-differential equations in Fourier coefficients. In its turn, the last system regarding unknown Fourier coefficients is recorded in the form of an integro-differential equation in the desired solution. Using relevant additional conditions of the auxiliary inverse boundary problem, we obtain a system of two nonlinear integral equations for defining unknown functions. Thus, the solution of the auxiliary inverse boundary problem comes down to the system of two nonlinear integro-differential equations in unknown functions. The specific Banach space is designed. Then, in the sphere made of the Banach space we with the help of contracted mapping prove the solvability of the nonlinear integro-differential equations set, which is a unique solution of the additional inverse boundary problem. Using the equivalence of problems, it is concluded about existence and uniqueness of a classical solution of the original problem.

The solutions of initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces

This paper discusses a boundary value problem of nonlinear fractional integrodifferential equations of order 1 < α ≤ 2 and 1 < β ≤ 2 and boundary conditions of the form x 0 = x 1 = D c β x 1 = D c β x 0 = 0 . Some new existence and uniqueness results are proposed by using the fixed point theory. In particular, we make use of the Banach contraction mapping principle and Krasnoselskii’s fixed point theorem under some weak conditions. Moreover, two illustrative examples are studied to support the results.

In this article, we consider a new class of fractional impulsive neutral stochastic functional integro-differential equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, fractional calculus, analytic α-resolvent operator and suitable fixed point theorems, we prove the existence of mild solutions and optimal mild solutions for these equations. Second, the existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. The results are obtained under weaker conditions in the sense of the fractional power arguments. Finally, an example is given for demonstration.

The Study aims in this paper to give and investigate the existence and uniqueness of mild solutions to nonlinear functional integrodifferential equations in Banach Spaces. the fixed point theorem, according to Sadovskii and sutible necessary conditions, are concepts consulted to obtain the results in the work