Abstract
Two collocation-based methods utilizing the novel Bessel polynomials (with positive coefficients) are developed for solving the non-linear Troesch’s problem. In the first approach, by expressing the unknown solution and its second derivative in terms of the Bessel matrix form along with some collocation points, the governing equation transforms into a non-linear algebraic matrix equation. In the second approach, the technique of quasi-linearization is first employed to linearize the model problem and, then, the first collocation method is applied to the sequence of linearized equations iteratively. In the latter approach, we require to solve a linear algebraic matrix equation in each iteration. Moreover, the error analysis of the Bessel series solution is established. In the end, numerical simulations and computational results are provided to illustrate the utility and applicability of the presented collocation approaches. Numerical comparisons with some existing available methods are performed to validate our results.
Highlights
Mathematical modeling through differential equations is of great importance in different branches of science and engineering with their ability to provide more realistic simulations to real life phenomenons
An efficient and reliable method based on the quasi-linearization technique combined with Bessel functions is proposed to obtain an accurate approximate series solution of Troesch’s model problem (1)
The basic ideas underlying the solutions of our model problems via quasi-linearization method (QLM) are briefly described, see [39–41] for more detailed discussions and applications
Summary
Mathematical modeling through differential equations is of great importance in different branches of science and engineering with their ability to provide more realistic simulations to real life phenomenons. 1− x sin t due to existence of a pole, which is located approximately at τs = ln(8/ψ0 (0))/α [4] These singularities hinder most of the numerical methods to capture the solutions of the Troesch’s problem, especially for large values of the parameter α. Tau method [20], the iterative method based on Green’s function and optimal homotopy analysis technique [21], the one-step hybrid block method [22], and the wavelet-homotopy analysis method [23] In this manuscript, an efficient and reliable method based on the quasi-linearization technique combined with Bessel functions is proposed to obtain an accurate approximate series solution of Troesch’s model problem (1). The direct Bessel collocation approach is developed for this model problem In the former method, we solve the original model via converting it into a sequence of linearized subproblems in conjunction with an iteration parameter r, which will be set at most 5.
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