Solvability of a fractional boundary value problem with fractional integral condition

Using Banach contraction principle and Leray-Schauder nonlinear alternative we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems for fractional differential equations with fractional integral condition, involving the Caputo fractional derivative. Some examples are given to illustrate our results.     

Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition

The main purpose of this paper is to present the existence results of solutions and positive solutions of nonlinear high-order fractional boundary value problems with integral boundary condition. By using the Banach fixed point theorem and the Krasnosel’skii fixed point theorem, we obtain the existence and uniqueness of real solution. By the Guo–Krasnosel’skii fixed point theorem on the cone, we obtain a desired result for guaranteeing the existence of positive solution. Several interesting examples relevant to the main results are also considered.     

A free boundary problem of diffusion equation with integral condition

We consider a free boundary problem of heat equation with integral condition on the unknown free boundary. Results of solution regularity and problem well-posedness are presented.     

On a semi-coercive problem in a porous journal bearing with cavitation problem as boundary condition and subject to an integral condition

We study a non-linear problem in pressure saturation modelling of a free boundary problem, arising in self-lubricating bearings, with Neumann boundary conditions for the pressure and a non-local constraint on the saturation variable, which indeed is a Lagrange multiplier. We prove an existence theorem by introducing an artificial time dependence and using the pseudo-characteristics discretization method and semi-coercive variational inequalities.     

On a boundary value problem with integral boundary conditions

We study the existence of positive solutions of second-order ordinary differential equations with integral boundary conditions. The result generalizes the conditions obtained in [1] for the existence of positive solutions.     

Stability of Rothe difference scheme for the reverse parabolic problem with integral boundary condition

In this paper, we study the approximation of reverse parabolic problem with integral boundary condition. The Rothe difference scheme for an approximate solution of reverse problem is discussed. We establish stability and coercive stability estimates for the solution of the Rothe difference scheme. In sequel, we investigate the first order of accuracy difference scheme for approximation of boundary value problem for multidimensional reverse parabolic equation and obtain stability estimates for its solution. Finally, we give numerical results together with an explanation on the realization in one- and two-dimensional test examples.     

An inverse coefficient problem for a quasilinear parabolic equation with periodic boundary and integral overdetermination condition

In this paper, the inverse problem of finding the time‐dependent coefficient of heat capacity together with the solution periodic boundary and integral overdetermination conditions is considered. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, and continuous dependence upon the data of the solution are shown. Some considerations on the numerical solution for this inverse problem are presented with an example. Copyright © 2014 John Wiley & Sons, Ltd.     

A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition

In this paper, we present a computer-assisted method that establishes the existence and local uniqueness of a stationary solution to the viscous Burgers’ equation. The problem formulation involves a left boundary condition and one integral boundary condition, which is a variation of a previous approach.     

A three-point boundary value problem with an integral condition for parabolic equations with the Bessel operator

In this paper, we study a three-point boundary value problem with an integral condition for a class of parabolic equation with Bessel operator. The existence and uniqueness of the solution in functional weighted Sobolev space are proved. The proof is based on two sided a priori estimates and the density of the range of the operator generated by the considered problem.     

Integral boundary problem in a layer

Translated from Matematicheskie Zametki, Vol. 53, No. 6, pp. 122–129, June, 1993.     

Existence of symmetric positive solutions for a boundary value problem with integral boundary conditions

This paper studies the existence of symmetric positive solutions for a second-order nonlinear ordinary differential equation with integral boundary conditions by applying the theory of fixed point index in cones. For the demonstration of the results, an illustrative example is presented.     

Solvability of a boundary-value problem with an integral boundary condition of the second kind for equations of odd order

We study the solvability of a boundary-value problem for equations of odd order subject to a boundary condition relating the values of the conormal derivative with those of an integral operator applied to the solution. We prove the existence and uniqueness theorems for regular solutions.     

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.     

Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition

We study non-negative solutions of the porous medium equationwith a source and a nonlinear flux boundary condition, u_t =(u^m)_xx + u^p in (0, ), x (0, T); – (u^m)_x (0, t) = u^q (0,t) for t (0, T); u (x, 0) = u₀ (x) in (0, ), where m > 1,p, q > 0 are parameters. For every fixed m we prove thatthere are two critical curves in the (p, q-plane: (i) the criticalexistence curve, separating the region where every solutionis global from the region where there exist blowing-up solutions,and (ii) the Fujita curve, separating a region of parametersin which all solutions blow up from a region where both globalin time solutions and blowing-up solutions exist. In the caseof blow up we find the blow-up rates, the blow-up sets and theblow-up profiles, showing that there is a phenomenon of asymptoticsimplification. If 2q < p + m the asymptotics are governedby the source term. On the other hand, if 2q > p + m theevolution close to blow up is ruled by the boundary flux. If2q = p + m both terms are of the same order.     

Boundary value problem for a parabolic-hyperbolic equation with a nonlocal integral condition

For a parabolic-hyperbolic equation with the heat and wave operators in a rectangular domain, we consider a problem with a nonlocal Samarksii-Ionkin condition. A criterion for the uniqueness of the solution is established by the spectral expansion method. The classical solution of the problem is constructed in the form of the sum of a biorthogonal series. The solution is proved to be stable with respect to the initial condition.     

A convex-concave problem with a nonlinear boundary condition

In this paper we study the existence of nontrivial solutions of the problem

     

Fractional order boundary value problem with integral boundary conditions involving Pettis integral

In this article, we investigate the existence of Pseudo solutions for some fractional order boundary value problem with integral boundary conditions in the Banach space of continuous function equipped with its weak topology. The class of such problems constitute a very interesting and important class of problems. They include two, three, multi-point and nonlocal boundary-value problems as special cases. In our investigation, the right hand side of the above problem is assumed to be Pettis integrable function. To encompass the full scope of this article, we give an example illustrating the main result.