Abstract We intend to numerically solve the famous two-dimensional Bratu's problem through developing a new iterative finite difference algorithm. The proposed scheme is capable of accurately approximating both branches of the solution of the considered problem. We first introduce a new transformation of Bratu's problem conserving the solution bifurcated behavior. Then, using Newton-Raphson-Kantorovich approximation in function space, we develop an iterative finite difference method yielding a simple algorithm for approximating the sequence of numerical solutions. We also perform a convergence analysis proving that our algorithm converges quadratically to the exact solution of the problem. Finally, we present numerical simulation results showing the capability of our algorithm to accurately compute the two-branches of the solution for the two-dimensional Bratu's problem.

中文翻译：

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二维布拉图问题的一个二分支数值解

摘要 我们打算通过开发一种新的迭代有限差分算法来数值求解著名的二维布拉图问题。所提出的方案能够准确地逼近所考虑问题的解决方案的两个分支。我们首先介绍了 Bratu 问题的新转换，以保存解分叉行为。然后，使用函数空间中的 Newton-Raphson-Kantorovich 逼近，我们开发了一种迭代有限差分方法，产生了一种用于逼近数值解序列的简单算法。我们还进行了收敛分析，证明我们的算法二次收敛到问题的精确解。最后，