The compact difference approach for the fourth‐order parabolic equations with the first Neumann boundary value conditions

Abstract

In this paper, the fourth‐order parabolic equation with the first Neumann boundary value conditions is concerned, where the values of the first and second spatial derivatives of the unknown function at the boundary are given. A compact difference scheme is established for this kind of problem by using the weighted average and order reduction methods. The difficulty lies in the challenges of handling the boundary conditions with high accuracy. The unique solvability, convergence and stability of the proposed compact difference scheme are proved by the energy method. Some novel techniques are introduced for the analysis. Then, the extension to a more general case with the reaction term is briefly explored. Finally, two numerical examples are numerically calculated to show the efficiency of the proposed numerical schemes.

Pointwise error estimate of the compact difference methods for the fourth‐order parabolic equations with the third Neumann boundary conditions

“…This motivates us to consider the compact difference method for solving the problem with the Neumann boundary conditions. The authors in [26] investigated the compact difference method for the first Neumann boundary value problem of the fourth‐order parabolic equations, where values of the first and second spatial derivatives of the unknown function are given at the boundary. The error estimate of the resultant compact scheme in $$$ {H}^1 $$$ seminorm was provided by the energy method.…”

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where $$$ f\left(x,t\right) $$$, $$$ q(t) $$$, $$$ \varphi (x) $$$, $$$ {\alpha}_i(t) $$$, $$$ {\beta}_i(t)\kern0.1em \left(i=2,3\right) $$$ are all given functions and $$$ {\varphi}^{\prime \prime }(0)={\alpha}_2(0) $$$, $$$ {\varphi}^{\prime \prime }(L)={\beta}_2(0) $$$, $$$ {\varphi}^{\prime \prime \prime }(0)={\alpha}_3(0) $$$, $$$ {\varphi}^{\prime \prime \prime }(L)={\beta}_3(0) $$$. To the best of our knowledge, many aspects need be reconsidered during the construction and analysis of numerical algorithms since the boundary conditions are different from those in [26]. For the problem (1.1)‐(1.4), a compact difference scheme will be established, analyzed, and numerically calculated.…”