This paper is devoted to find the numerical solutions of one dimensional general nonlinear system of third-order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual method. We derive mathematical formulations in matrix form, in details, by exploiting Bernstein polynomials as basis functions. A reasonable accuracy is found when the proposed method is used on few examples. At the end of the study, a comparison is made between the approximate and exact solutions, and also with the solutions of the existing methods. Our results converge monotonically to the exact solutions. In addition, we show that the the derived formulations may be applicable by reducing higher order complicated BVP into a lower order system of BVPs, and the performance of the numerical solutions is satisfactory.

翻译：本文致力于利用 Galerkin 加权残量法求解一对函数的一维一般非线性三阶边值问题系统的数值解。通过采用 Bernstein 多项式作为基函数，我们详细推导了矩阵形式的数学公式。在若干算例中应用所提方法时，获得了合理的精度。研究最后，将近似解与精确解进行了比较，并同现有方法的解进行了对比。我们的结果单调收敛于精确解。此外，我们证明，通过将高阶复杂边值问题降阶为低阶边值问题系统，所推导的公式可能具有适用性，且数值解的性能令人满意。