If there is a jump discontinuity present in the forcing term of a boundary value problem (BVP), the nonstandard finite difference (NSFD) and finite difference (FD) methods do not approximate the solutions very well. Here we use fuzzy transforms (FTs) and derive fuzzy transformed NSFD schemes that are referred to as non-standard fuzzy transform methods (NFTMs). The convergence of the derived NFTMs is established. Numerical solutions of Lane–Emden type equations are obtained using NFTMs. We show that NFTMs provide better results than NSFD and FD methods when the forcing term has a jump discontinuity. Even for large jumps, the NFTMs provide more accurate results than the other methods.

中文翻译：

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修正的Mickens型NSFD格式在Lane-Emden方程中的应用

如果在边值问题（BVP）的强制项中存在跳跃不连续性，则非标准有限差分（NSFD）和有限差分（FD）方法不能很好地近似求解。在这里，我们使用模糊变换（FTs）并得出模糊变换的NSFD方案，这些方案称为非标准模糊变换方法（NFTM）。建立了派生的NFTM的收敛性。使用NFTM获得Lane-Emden型方程的数值解。我们显示，当强迫项具有跳跃间断时，NFTM比NSFD和FD方法提供更好的结果。即使对于较大的跳跃，NFTM也比其他方法提供更准确的结果。