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Wavelet solution of a strongly nonlinear Lane–Emden equation

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Abstract

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, i.e., orthogonal polynomial wavelet method (OPWM). We also discuss the convergence of OPWM as a particular case. We discuss multiresolution analysis for wavelets generated by orthogonal polynomials, e.g., Hermite, Legendre, Chebyshev, Laguerre, and Gegenbauer. Then we use these orthogonal polynomial 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.

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Authors and Affiliations

  1. Faculty of Mathematics, University of Vienna, Vienna, Austria

    Diksha Tiwari

  2. Department of Mathematics, IIT Patna, Patna, Bihar, 801106, India

    Amit K. Verma

  3. Engineering School (DEIM), University of Tuscia, Largo dell’Universit a, 01100, Viterbo, Italy

    Carlo Cattani

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  1. Diksha Tiwari
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  2. Amit K. Verma
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Correspondence to Amit K. Verma.

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Tiwari, D., Verma, A.K. & Cattani, C. Wavelet solution of a strongly nonlinear Lane–Emden equation. J Math Chem 60, 2054–2080 (2022). https://doi.org/10.1007/s10910-022-01401-3

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