最优同调分析法与格林函数用于一类非局部椭圆边值问题
Optimal homotopy analysis method with Green's function for a class of nonlocal elliptic boundary value problems
AI总结：
本文提出结合格林函数的最优同调分析法求解非局部椭圆问题，通过转换为积分方程并利用收敛控制参数c_0，验证了方法的收敛性和准确性，无需额外计算未知常数，具有良好的收敛性和灵活性。
摘要：
在本文中，我们提出一种结合格林函数技术的最优同调分析法（OHAM），用于获取非局部椭圆问题的准确数值解。我们首先将非局部边界值问题转换为等价的积分方程，然后使用具有收敛控制参数c_0的OHAM。为了证明OHAM方法的收敛性和准确性，我们比较了OHAM与Adomian分解法（ADM）结合格林函数的结果。数值实验证实了该方法的可靠性，因为它能够处理此类非局部椭圆微分方程，而无需施加可能改变解物理结构的限制假设。我们还讨论了所提出方法的收敛性和误差分析。总结：(i) 该方法不同于ADM和VIM[1]，无需额外计算未知常数；(ii) 收敛性保证；(iii) 解的初始猜测选择具有灵活性；(iv) 一种有用的解析工具，用于研究一类非局部椭圆边界值问题。
英文摘要：
In this paper, we present the optimal homotopy analysis method (OHAM) with Green's function technique to acquire accurate numerical solutions for the nonlocal elliptic problems. We first transform the nonlocal boundary value problems into an equivalent integral equation, and then use an OHAM with convergence control parameter $c_0$. To demonstrate convergence and accuracy characteristics of the OHAM method, we compare the OHAM and Adomian decomposition method (ADM) with Green's function. The numerical experiments confirm the reliability of the approach as it handles such nonlocal elliptic differential equations without imposing limiting assumptions that could change the physical structure of the solution. We also discuss the convergence and error analysis of proposed method. In summary: $(i)$ the present approach does not require any additional computational work for unknown constants unlike ADM and VIM \cite{khuri2014variational} $(ii)$ guarantee of convergence $(iii)$ flexibility on choice of initial guess of solution and $(iv)$ useful analytic tool to investigate a class of nonlocal elliptic boundary value problems.
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