Study of some linear and nonlinear heat problem with nonlocal nonlinear boundary conditions
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Date
2021
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Université de Larbi Ben M’hidi-Oum El Bouaghi
Abstract
Partial differential equations with nonlinear nonlocal boundary conditions occur increasingly in many fields of science and engineering.
In this work, we focus to study the existence and uniqueness of class of linear and nonlinear parabolic problems with nonlinear nonlocal second kind integral conditions by using the energy inequality method and an iterative process based on the results of the a linear case.
The presence of integral conditions complicates the use of standared numerical methodes; so fourth-order compact finte diference scheme is developed to solve the diffusion equation with nonlinear nonlocal integral boundary conditions of second type. Where, the proposed scheme is derived by combining a fourth-order compact finite difference formula in space and a backward differentiation for the time dirivative term. Also, the non linear terms are linearized by using Taylor expansion. Therefore, numerical examples are provided to verify the accuracy and efficiency of our proposed method.
Finally, we also cared to the theorical and numerical finite-time explosion of the semilinear problem.
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Keywords
Parabolic equation, Energy inequality, Nonlinear parabolic, Numerical solution