一种基于正交多项式小波的广义新方法及其在Lane-Emden方程中的应用
A generalized novel approach based on orthonormal polynomial wavelets with an application to Lane-Emden equation
摘要：
由于微分方程中的系数在奇点附近爆炸，因此捕获任何非线性SBVP的奇点附近的解是具有挑战性的。在这篇文章中，我们的目的是建立一个通用的方法，基于正交多项式的小波。我们讨论了由正交多项式生成的小波的多分辨率分析，例如，Hermite，Legendre，Chebyshev，Laguerre，and Gegenbauer.然后，我们使用这些小波求解非线性SBVP。这些小波可以方便有效地处理奇异性。为了处理非线性，我们使用牛顿拟线性化和牛顿-拉夫森方法。为了显示所提出的方法的重要性和准确性，我们解决了莱恩-埃姆登类型的问题，并比较计算的解决方案与已知的解决方案。随着分辨率的增加，计算解收敛到精确解或已知解。我们观察到，所提出的技术表现良好的一类莱恩-埃姆登型BVP。由于本文处理的是奇异性、非线性的问题，并采用不同的小波对结果进行了比较。
英文摘要：
Capturing solution near the singular point of any nonlinear SBVPs is challenging because coefficients involved in the differential equation blow up near singularities. In this article, we aim to construct a general method based on orthogonal polynomials as wavelets. We discuss multiresolution analysis for wavelets generated by orthogonal polynomials, e.g., Hermite, Legendre, Chebyshev, Laguerre, and Gegenbauer. Then we use these wavelets for solving nonlinear SBVPs. These wavelets can deal with singularities easily and efficiently. To deal with the nonlinearity, we use both Newton's quasilinearization and the Newton-Raphson method. To show the importance and accuracy of the proposed methods, we solve the Lane-Emden type of problems and compare the computed solutions with the known solutions. As the resolution is increased the computed solutions converge to exact solutions or known solutions. We observe that the proposed technique performs well on a class of Lane-Emden type BVPs. As the paper deals with singularity, non-linearity significantly and different wavelets are used to compare the results.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
