On the solvability of a class of singular parabolic equations with nonlocal boundary conditions in nonclassical function spaces

The aim of this paper is to prove the existence, uniqueness, and continuous dependence upon the data of a generalized solution for certain singular parabolic equations with initial and nonlocal boundary conditions. The proof is based on an a priori estimate established in nonclassical function spaces, and on the density of the range of the operator corresponding to the abstract formulation of the considered problem

Solvability the telegraph equation with purely integral conditions

In this paper a numerical technique is developed for the one-dimensional telegraph equation. We prove the existence, uniqueness, and continuous dependence upon the data of solution to a telegraph equation with purely integral conditions. The proofs are based on a priori estimates and Laplace transform method. Finally, we obtain the solution by using a simple and efficient algorithm for numerical solution.Publisher's Versio

A mixed problem with only integral boundary conditions for a hyperbolic equation

We investigate an initial boundary value problem for a second-order hyperbolic equation with only integral conditions. We show the existence, uniqueness, and continuous dependence of a strongly generalized solution. The proof is based on an energy inequality established in a nonclassical function space, and on the density of the range of the operator associated to the abstract formulation of the studied problem by introducing special smoothing operators

Solvability the telegraph equation with purely integral conditions

In this paper a numerical technique is developed for the one-dimensional telegraph equation, we prove the existence, uniqueness, and continuous dependence upon the data of solution to a telegraph equation with purely integral conditions. The proofs are based on a priori estimates and Laplace transform method. Finally, we obtain the solution by using a simple and efficient algorithm for numerical solution.Publisher's Versio

On three-point boundary value problem with a weighted integral condition for a class of singular parabolic equations

We deal with a three point boundary value problem for a class of singular parabolic equations with a weighted integral condition in place of one of standard boundary conditions. We will first establish an a priori estimate in weighted spaces. Then, we prove the existence, uniqueness, and continuous dependence of a strong solution

On the solvability of a class of reaction-diffusion systems

We deal with a class of parabolic reaction-diffusion systems. We use an iterative process based on results obtained for a linearized problem, then we derive some a priori estimates to establish the existence, uniqueness, and continuous dependence of the weak solution for a class of quasilinear systems

Study of Solution for a Parabolic Integrodifferential Equation with the Second Kind Integral Condition

In this paper, we establish sufficient conditions for the existence, uniqueness and numerical solution for a parabolic integrodifferential equation with the second kind integral condition. The existence, uniqueness of a strong solution for the linear problem based on a priori estimate “energy inequality” and transformation of the linear problem to linear first-order ordinary differential equation with second member. Then by using a priori estimate and applying an iterative process based on results obtained for the linear problem, we prove the existence, uniqueness of the weak generalized solution of the integrodifferential prob- lem. Also we have developed an efficient numerical scheme, which uses temporary problems with standard boundary conditions. A suitable combination of the auxiliary solutions defines an approximate solution to the original nonlocal problem, the algebraic matrices obtained after the full discretization are tridiagonal, then the solution is obtained by using the Thomas algorithm. Some numerical results are reported to show the efficiency and accuracy of the scheme

On initial boundary value problem with Dirichlet integral conditions for a hyperbolic equation with the Bessel operator

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem

On the weak solution of a three-point boundary value problem for a class of parabolic equations with energy specification

This paper deals with weak solution in weighted Sobolev spaces, of three-point boundary value problems which combine Dirichlet and integral conditions, for linear and quasilinear parabolic equations in a domain with curved lateral boundaries. We, firstly, prove the existence, uniqueness, and continuous dependence of the solution for the linear equation. Next, analogous results are established for the quasilinear problem, using an iterative process based on results obtained for the linear problem

On the solvability of parabolic and hyperbolic problems with a boundary integral condition

We prove the existence, uniqueness, and the continuous dependence of a generalized solution upon the data of certain parabolic and hyperbolic equations with a boundary integral condition. The proof uses a functional analysis method based on a priori estimates established in nonclassical function spaces, and on the density of the range of the linear operator associated to the abstract formulation of the studied problem