This article aims to study the existence and uniqueness of a weak solution for a singular and degenerate nonlinear parabolic equation with a generalized nonlinear integral condition of the second type. The proof of the existence and uniqueness of the weak solution to such a problem will be proceeded with in three steps. In the same regard, the solvability of the linear case of the problem at hand will be handled with the use of the Faedo-Galerkin method, a priori estimate, and by imposing some nonlinear conditions of the second kind.

This paper is devoted to the study of linear nonlocal problems Dirichlet condition and Neumann condition modeling integration condition a second class of a class of linear reaction-diffusion equations.We show the existence and uniqueness of weak solutions to problems Fadeo-Galarkin method developed to circumvent the resulting complexities due to the existence of integration conditions.We also seek numerical solutions using finite difference techniques.

Under the consideration that the non-local condition has a spatial impact to the study of the boundary values problems, we present a study of the existence and uniqueness of weak solution for nonlinear parabolic Bessel problem with Neumann integral conditions, in addition to part devoted to the proof of the finite time blow up solutions. Actually, in the case p >= 1, sufficient conditions of blow up of solutions can be established by Kaplan's method backed by the numerical results.

In this work, we prove the existence, uniqueness, and continuous dependence of generalized solution of a nonlinear reaction-diffusion problem with only integral terms in the boundaries, by using the finite element method.Also we have developed an efficient numerical finite difference schemes. Some numerical results are reported to show the efficiency and accuracy of the scheme.

In this paper we present a class of parabolic equation with nonlinear nonlocal conditions of second type where we show two part of this study the theoretical part we prove the existence and uniqueness of the solution by energy inequality method. Then the numerical part where we study the consistence and stability of solution.

In this article, Modified Backward Euler Scheme is developed to solve the diffusion equation subject to nonlinear nonlocal boundary conditions. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time derivative term. Nonlinear terms in boundary conditions are linearized by Taylor expansion. Numerical examples are provided to verify the accuracy and efficiency of our proposed method.

In this paper, explicit and implicit Crandall’s formulas are applied for finding the solution of the one-dimensional heat equation with nonlinear nonlocal boundary conditions. The integrals in the boundary equations are approximated by the composite Simpson quadrature rule. Here nonlinear terms are approximated by Richtmyer’s linearization method. Finally, some numerical examples are given to show the effectiveness of the proposed method.

In this article, Modified Backward Euler Scheme is developed to solve the diffusion equation subject to nonlinear nonlocal boundary conditions. The proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differ- entiation for the time derivative term. Nonlinear terms in boundary conditions are linearized by Taylor expansion. Numerical examples are provided to verify the accuracy and efficiency of our proposed method.

In this paper, we will study practical problems of optimal control governed by differential equations with partial derivatives. In the framework of this work, we will deal with an optimal control problem applied to the heat equation. This problem has many applications in the field of physics and in automation.

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 problem. 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.

A nonlinear parabolic problem with a nonlocal boundary condition is studied. We prove the existence of a solution for a monotonically increasing and Lipschitz continuous nonlinearity. The approximation method is based on Rothe's method. The solution on each time step is obtained by iterations, convergence of which is shown using a fixed-point argument. The space discretization relies on FEM. Theoretical results are supported by numerical experiments.

A semilinear reaction-diffusion problem with a nonlocal boundary condition is studied. This paper presents a new and very easy implementable numerical algorithm for computations. This is based on a suitable linearization in time and on the principle of linear superposition. Any method for the space discretization (FEM was taken in this analysis) can be chosen. The derived algorithm is implicit and it does not need any iteration scheme to get a solution with the nonlocal boundary condition. Stability analysis has been performed and the optimal error estimates have been derived. Numerical results have been compared with other known techniques.