Nonlocal Integral Conditions Research Articles

On solvability of the integrodifferential hyperbolic equation with purely nonlocal conditions

In this study, we prove the existence, uniqueness, and continuous dependence upon the data of solution to integro-differential hyperbolic equation with purely nonlocal (integral) conditions. The proofs are based on a priori estimates and Laplace transform method. Finally, we obtain the solution using a numerical technique (Stehfest algorithm) by inverting the Laplace transform.

Solvability of a boundary value problem for a pseudoparabolic equation with nonlocal integral conditions

By using the method of continuation with respect to a parameter and a priori estimates, we prove the unique regular solvability of a boundary value problem with nonlocal integral boundary conditions for a one-dimensional pseudoparabolic equation.

Boundary-Value Problem with Nonlocal Integral Condition for Mixed-Type Equations with Degeneracy on the Transition Line

Для уравнения эллиптико-гиперболического типа решена краевая задача с нелокальным условием Самарского-Ионкина в прямоугольной области. Методом спектрального анализа установлен критерий единственности и доказана теорема существования решения поставленной задачи. Решение задачи построено в виде суммы биортогонального ряда. Библиография: 10 названий.

Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions

Abstract In this paper, we study the existence and uniqueness of solution for a problem consisting of a sequential nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions. A variety of fixed point theorems, such as Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder degree theory, are used. Examples illustrating the obtained results are also presented. MSC:26A33, 34A08, 34B10.

Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations

In this paper, we introduce a new class of boundary value problems consisting of a fractional differential equation of Riemann-Liouville type, D q R L x(t)=f(t,x(t)), t∈[0,T], subject to the Hadamard fractional integral conditions x(0)=0, x(T)= ∑ i = 1 n α i H I p i x( η i ). Existence and uniqueness results are obtained by using a variety of fixed point theorems. Examples illustrating the results obtained are also presented.MSC: 34A08, 34B15.

Existence of solution of a Coupled system of differential equation with nonlocal conditions

In this paper, we study the existence of at least one solution of the coupled system of differential equations with nonlocal conditions. Also, a coupled system of differential equations with the nonlocal integral conditions will be considered.

Classical solution of a problem with an integral condition for the one-dimensional wave equation

We find a closed-form classical solution of the homogeneous wave equation with Cauchy conditions, a boundary condition on the lateral boundary, and a nonlocal integral condition involving the values of the solution at interior points of the domain. A classical solution is understood as a function that is defined everywhere in the closure of the domain and has all classical derivatives occurring in the equation and conditions of the problem. The derivatives are defined via the limit values of finite-difference ratios of the function and corresponding increments of the arguments.

Identifying a time-dependent heat source with nonlocal boundary and overdetermination conditions by the variational iteration method

Purpose – The purpose of this paper is to investigate the inverse problem of determining a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. Design/methodology/approach – The variational iteration method (VIM) is employed as a numerical technique to develop numerical solution. A numerical example is presented to illustrate the advantages of the method. Findings – Using this method, we obtain the exact solution of this problem. Whether or not there is a noisy overdetermination data, our numerical results are stable. Thus the VIM is suitable for finding the approximation solution of the problem. Originality/value – This method is based on the use of Lagrange multipliers for the identification of optimal values of parameters in a functional and gives rapidly convergent successive approximations of the exact solution if such a solution exists.

Numerical solution of time-dependent diffusion equations with nonlocal boundary conditions via a fast matrix approach

Abstract This article contributes a matrix approach by using Taylor approximation to obtain the numerical solution of one-dimensional time-dependent parabolic partial differential equations (PDEs) subject to nonlocal boundary integral conditions. We first impose the initial and boundary conditions to the main problems and then reach to the associated integro-PDEs. By using operational matrices and also the completeness of the monomials basis, the obtained integro-PDEs will be reduced to the generalized Sylvester equations. For solving these algebraic systems, we apply a famous technique in Krylov subspace iterative methods. A numerical example is considered to show the efficiency of the proposed idea.

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C 0-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.

Nonlocal Parabolic Problem with Degeneration

We study the problem for a second-order linear parabolic equation with nonlocal integral condition in the time variable and power singularities in the coefficients of any order with respect to the time and space variables. By using the maximum principle and a priori estimates, we establish the existence and uniqueness of the solution of this problem in Holder spaces with power weights.

Determination of a time-dependent coefficient in the bioheat equation

In this paper, the identification of the time-dependent blood perfusion coefficient in the bioheat equation is considered as an inverse heat source problem with nonlocal boundary and integral energy over-determination conditions. The boundary element method (BEM) based on using the fundamental solution for the heat equation is employed, together with either the second-order Tikhonov regularization combined with finite differences, or with a smoothing spline regularization technique for computing the first-order derivative of a noisy function. A couple of benchmark numerical examples are presented to verify the accuracy and stability of the solution.

Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions

The development of numerical methods for the solution of partial differential equations (PDEs) with nonlocal boundary conditions is important, since such type of problems arise as mathematical models of various real-world processes. We use radial basis function (RBF) collocation technique for the solution of a multidimensional linear elliptic equation with classical Dirichlet boundary condition and nonlocal integral conditions. RBF-based meshless methods are easily implemented and efficient, especially for multidimensional problems formulated on complexly shaped domains. In this paper, properties of the method are investigated by studying two- and three-dimensional test examples with manufactured solutions. We analyze the influence of the RBF shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix.

An Inverse Coefficient Problem for a Parabolic Equation under Nonlocal Boundary and Integral Overdetermination Conditions

This paper investigates the inverse problem of simultaneously determining the time-dependent thermal diffusivity and the temperature distribution in a parabolic equation in the case of nonlocal boundary conditions containing a real parameter and integral overdetermination conditions. Under some consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the classical solution are shown by using the generalized Fourier method.

Nonlinear Langevin Equation of Hadamard-Caputo Type Fractional Derivatives with Nonlocal Fractional Integral Conditions

We study existence and uniqueness of solutions for a problem consisting of nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions. A variety of fixed point theorems are used, such as Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative, and Leray-Schauder’s degree theory. Enlightening examples illustrating the obtained results are also presented.

КЛАССИЧЕСКОЕ РЕШЕНИЕ ЗАДАЧИ С ИНТЕГРАЛЬНЫМ УСЛОВИЕМ ДЛЯ ОДНОМЕРНОГО ВОЛНОВОГО УРАВНЕНИЯ

We find a closed-form classical solution of the homogeneous biwave equation with Cauchy conditions, a boundary condition on the lateral boundary, and a nonlocal integral condition involving the values of the solution at interior points of the domain. A classical solution is understood as a function that is defined everywhere in the closure of the domain and has all classical derivatives occurring in the equation and conditions of the problem.

Investigation of the spectrum of the Sturm–Liouville problem with a nonlocal integral condition

This paper presents some new results on the spectrum for the second order dif-ferential problem with one integral type nonlocal boundary condition (NBC). We investigate how the spectrum of this problem depends on the integral nonlocal boundary condition pa-rameters γ, ξ and the symmetric interval in the integral. Some new results are given on the complex spectra of this problem. Many results are presented as graphs of real and complex characteristic functions.

Numerical solution of nonlinear elliptic equation with nonlocal condition

Two iterative methods are considered for the system of difference equations approximating two-dimensional nonlinear elliptic equation with the nonlocal integral condition. Motivation and possible applications of the problem present in the paper coincide with the small volume problems in hydrodynamics. The differential problem considered in the article is some generalization of the boundary value problem for minimal surface equation.

APPROXIMATE SOLUTION TO A MULTI-POINT BOUNDARY VALUE PROBLEM INVOLVING NONLOCAL INTEGRAL CONDITIONS BY REPRODUCING KERNEL METHOD

In this work, we investigate a sequence of approximations converging to the existing unique solution of a multi-point boundary value problem(BVP) given by a linear fourth-order ordinary differential equation with variable coeffcients involving nonlocal integral conditions by using reproducing kernel method(RKM). Obtaining the reproducing kernel of the reproducing kernel space by using the original conditions given directly by RKM may be troublesome and may introduce computational costs. Therefore, in these cases, initially considering more admissible conditions which will allow the reproducing kernel to be computed more easily than the original ones and then taking into account the original conditions lead us to satisfactory results. This analysis is illustrated by a numerical example. The results demonstrate that the method is still quite accurate and effective for the cases with both derivative and integral conditions even if the accuracy is less compared to the cases with just derivative conditions.

An inverse time-dependent source problem for the heat equation

In this paper the inverse problem of finding the time-dependent heat source together with the temperature solution of heat equation with nonlocal boundary and integral additional conditions is investigated. The existence, uniqueness and continuous dependence of the solution of the inverse problem upon the data are established. Furthermore, a boundary element method combined with the Tikhonov regularization of various orders is developed in order to obtain a stable solution. Numerical results are presented and discussed.