In this paper, we propose a computational technique based on a combination of optimal homotopy analysis method (OHAM) and an iterative finite difference method (FDM) for a class of derivative-dependent doubly singular boundary value problems:

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The principal idea of this approach is to decompose the domain of the problem \(D=[0,1]\) into two subdomains as \(D=D_{1}\cup D_{2}=[0,\gamma ]\cup [\gamma ,1]\) (\(\gamma \) is the vicinity of the singularity). In the first domain \(D_{1},\) we use OHAM to overcome the singularity behaviour at \(x=0\). In the second domain \(D_{2}\), a FDM is designed for solving the resulting regular boundary value problem. Convergence analysis of the method is carried out. Three nonlinear examples are considered to demonstrate the performance and accuracy of the proposed method. It is shown that the computational order of convergence of the FDM is two.

在本文中，我们针对一类依赖于导数的双奇异边值问题，提出了一种基于最优同伦分析方法（OHAM）和迭代有限差分方法（FDM）相结合的计算技术：

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这种方法的主要思想是将问题\（D = [0,1] \）分解为两个子域，即\（D = D_ {1} \ cup D_ {2} = [0，\ gamma] \ cup [\ gamma，1] \）（\（\ gamma \）是奇点附近）。在第一个域\（D_ {1}，\）中，我们使用OHAM来克服\（x = 0 \）处的奇异行为。在第二个域\（D_ {2} \）中，设计了FDM来解决所得的规则边值问题。对该方法进行了收敛性分析。考虑了三个非线性示例，以证明该方法的性能和准确性。结果表明，FDM收敛的计算顺序为二。