https://doi.org/10.1140/epjs/s11734-025-01933-0
Regular Article
Newly generated path graph matching-polynomial matrix of integration to solve initial-value, finite-value and infinite-boundary-value problems
1
Department of Studies and Research in Mathematics, Tumkur University, Jnanasiri Campus, Bidarakatte, 572118, Tumakuru, Karnataka, India
2
Department of Mechanical Engineering, Birla Institute of Technology and Science Pilani, Pilani Campus, Vidya Vihar, 333031, Pilani, Rajasthan, India
3
Department of Power Engineering, Jadavpur University, Salt Lake, 700106, Kolkata, India
Received:
19
June
2025
Accepted:
5
September
2025
Published online:
24
September
2025
This work introduces a new class of matching polynomials from path graphs, embedded into a closed-form operational matrix of integration for solving differential equations. Unlike conventional spectral methods, the approach avoids orthogonalization, uses symbolic–numeric computation, and reuses a precomputed integration matrix to reduce computational cost. The method effectively handles both linear and nonlinear differential equations, offering flexibility in addressing diverse boundary conditions, including transformed infinite boundaries. The computational efficiency is further enhanced by the reuse of precomputed integration matrices, leading to reduced memory usage and faster runtimes. The method is applied to initial-value, finite-boundary, and transformed-infinite boundary problems, with examples from Williamson fluid flow and a nonlinear atmospheric model. Comparisons with exact solutions, numerical solvers, and previous wavelet-based approaches confirm spectral-like accuracy, stability, and efficiency, demonstrating its versatility for nonlinear dynamical systems across diverse scientific domains. Furthermore, the method is applied to climate modeling, illustrating its broader applicability to problems in environmental science and public health. The method robustness, fast convergence, and ease of implementation make it an effective tool for solving complex real-world differential equations across diverse scientific and engineering disciplines.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
