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A stable higher-order numerical method for solving a system of third-order singular Emden-Fowler type equations

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Abstract

This paper proposes a new higher-order numerical method based on a difference scheme with uniform steps to solve a strongly nonlinear system of third-order singular Emden-Fowler-type equations. These problems are challenging to solve because of their singularity or strong nonlinearity. To handle the singularity of the problem, we approximate the derivatives at the endpoints and develop a new difference scheme. This scheme provides a system of nonlinear equations solved by an iterative method. Also, we mathematically establish the method’s stability, consistency, and convergence analysis using a matrix analysis approach. We also verify the presented technique’s efficiency, accuracy and applicability by solving different examples from the literature. We also show that the theoretical order of the technique is consistent with the numerical convergence rates. Additionally, our method easily achieves higher-order accuracy with minimal grid points, unlike most methods that typically require modifying the equation into an equivalent integral equation or using L’Hospital’s rule to remove singularities, resulting in lower-order accuracy approaches.

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  1. Department of Mathematics, Birla Institute of Technology Mesra, Ranchi, 835215, India

    Nirupam Sahoo & Randhir Singh

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  1. Nirupam Sahoo
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  2. Randhir Singh
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N.S.: Contributed to formulation, Methodology, Visualization, Investigation, Programming, Writing - Original Draft. R.S.: Contributed to formulation, Methodology, Investigation, Writing- Review and Editing.

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Correspondence to Randhir Singh.

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Appendix

Appendix

Here, we present the system of nonlinear algebraic equations (at least 6 equations), solved by the Newton–Raphson method for Example 1:

for \(i=1\)

$$\begin{aligned} & \frac{740449}{1492992} z_{1,2}^9 z_{2,2}^2 - \frac{64}{5} z_{1,2} - \frac{429}{20} z_{1,3} + \frac{662}{15} z_{1,4} \\ & \qquad - \frac{431}{20} z_{1,5} + 6 z_{1,6} - \frac{3}{4} z_{1,7} + \frac{77}{12} = 0,\\ & \frac{10909}{5832} z_{1,3}^9 z_{2,3}^2 - 3 z_{1,3} + 16 z_{1,2} - \frac{232}{3} z_{1,4} + 83 z_{1,5} \\ & \qquad - 24 z_{1,6} + 3 z_{1,7} + \frac{7}{3} = 0,\\ & \frac{7209}{2048} z_{1,4}^9 z_{2,4}^2 - \frac{302}{5} z_{1,4} + \frac{432}{5} z_{1,2} + \frac{27}{20} z_{1,3} - \frac{567}{20} z_{1,5} \\ & \qquad + 54 z_{1,6} - \frac{27}{4} z_{1,7} - \frac{185}{4} = 0,\\ & \frac{2840}{729} z_{1,5}^9 z_{2,5}^2 - 16 z_{1,5} - 1376 z_{1,2} + 1644 z_{1,3} \\ & \qquad - \frac{2432}{3} z_{1,4} + 96 z_{1,6} + 12 z_{1,7} + \frac{1352}{3} = 0,\\ & \frac{153025}{1492992} z_{1,6}^9 z_{2,6}^2 + 426 z_{1,6} +16420 z_{1,2} - \frac{91281}{4} z_{1,3} + \frac{50246}{3} z_{1,4} \\ & \qquad - \frac{24667}{4} z_{1,5} + \frac{1125}{4} z_{1,7} - \frac{58667}{12} = 0,\\ & 280809 z_{1,3} - 193824 z_{1,2} - 223832 z_{1,4} + 105075 z_{1,5} - 27144 z_{1,6} + 3087 z_{1,7} \\ & \qquad - \frac{99}{8} z_{1,7}^9 z_{2,7}^2 + 55829 = 0, \end{aligned}$$

for \(i=2\)

$$\begin{aligned} & \frac{662}{15} z_{2,4} - \frac{429}{20} z_{2,3} - \frac{64}{5} z_{2,2} - \frac{431}{20} z_{2,5} +6 z_{2,6} \\ & \qquad - \frac{3}{4} z_{2,7} - \frac{749093}{1492992} z_{1,2}^7 z_{2,2}^2 + \frac{77}{12} =0,\\ & 16 z_{2,2} - 3 z_{2,3} - \frac{232}{3} z_{2,4} + 83 z_{2,5} - 24 z_{2,6} + 3 z_{2,7} - \frac{11993}{5832} z_{1,3}^7 z_{2,3}^2 + \frac{7}{3} =0,\\ & \frac{432}{5} z_{2,2} + \frac{27}{20} z_{2,3} - \frac{302}{5} z_{2,4} - \frac{567}{20} z_{2,5} + 54 z_{2,6} - \frac{27}{4} z_{2,7}\\ & \qquad - \frac{10125}{2048} z_{1,4}^7 z_{2,4}^2 - \frac{185}{4} =0,\\ & 1644 z_{2,3} - 1376 z_{2,2} - \frac{2432}{3} z_{2,4} - 16 z_{2,5} + 96 z_{2,6} + 12 z_{2,7}\\ & \qquad - \frac{7288}{729} z_{1,5}^7 z_{2,5}^2 + \frac{1352}{3} =0,\\ & 16420 z_{2,2} - \frac{91281}{4} z_{2,3} + \frac{50246}{3} z_{2,4} - \frac{24667}{4} z_{2,5} + 426 z_{2,6} + \frac{1125}{4} z_{2,7} \\ & \qquad - \frac{28715525}{1492992} z_{1,6}^7 z_{2,6}^2 - \frac{58667}{12} =0,\\ & 280809 z_{2,3} - 193824 z_{2,2} - 223832 z_{2,4} + 105075 z_{2,5} - 27144 z_{2,6} + 3087 z_{2,7} \\ & \qquad - \frac{297}{8} z_{1,7}^7 z_{2,7}^2 + 55829 =0. \end{aligned}$$

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Sahoo, N., Singh, R. A stable higher-order numerical method for solving a system of third-order singular Emden-Fowler type equations. J. Appl. Math. Comput. 71, 387–414 (2025). https://doi.org/10.1007/s12190-024-02233-x

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