Numerical solution of time-dependent Emden-Fowler equations using bivariate spectral collocation method on overlapping grids

Abstract

<jats:title>Abstract</jats:title><jats:p>In this work, we present a new modification to the bivariate spectral collocation method in solving Emden-Fowler equations. The novelty of the modified approach is the use of overlapping grids when applying the Chebyshev spectral collocation method. In the case of nonlinear partial differential equations, the quasilinearisation method is used to linearize the equation. The multi-domain technique is applied in both space and time intervals, which are both decomposed into overlapping subintervals. The spectral collocation method is then employed in the discretization of the iterative scheme to give a matrix system to be solved simultaneously across the overlapping subintervals. Several test examples are considered to demonstrate the general performance of the numerical technique in terms of efficiency and accuracy. The numerical solutions are matched against exact solutions to confirm the accuracy and convergence of the method. The error bound theorems and proofs have been considered to emphasize on the benefits of the method. The use of an overlapping grid gives a matrix system with less dense matrices that can be inverted in a computationally efficient manner. Thus, implementing the spectral collocation method on overlapping grids improves the computational time and accuracy. Furthermore, few grid points in each subinterval are required to achieve stable and accurate results. The approximate solutions are established to be in excellent agreement with the exact analytical solutions.</jats:p>

Overlapping grid spectral collocation approach for electrical MHD bioconvection Darcy–Forchheimer flow of a Carreau–Yasuda nanoliquid over a periodically accelerating surface

“…For the derivation of Equation ( 24), readers are referred to studies by Mkhatshwa et al 61 Similarly, the spatial domain ξ ξ ξ ,…”

“…In the current work, MD-BSLLM is improved through grid-manipulation which involves the use of overlapping grid ideas in space intervals when discretizing the solution domain. Overlapping grid procedures have been confirmed in some studies 43,61 to improve accuracy, convergence, computational efficiency, and stability of the bivariate spectral quasilinearization method. The overlapping MD-BSLLM approach can easily be developed and implemented in scientific computer software.…”

MHD mixed convective radiative flow of Eyring‐Powell fluid over an oscillatory stretching sheet using bivariate spectral method on overlapping grids

“…To achieve such maximum efficiency as well as improve the production process, we have considered one of the important non‐Newtonian fluid with some more useful factors and valuable applications. The novelty of the present study also lies in the use of efficient bivariate spectral quasilinearization method (BSQLM) on overlapping grids 38 in solving the dimensionless flow equations. The single‐domain BSQLM was introduced by Motsa et al 39 and used in solving nonlinear evolution partial differential equations (PDEs).…”

“…Mkhatshwa et al 48 and Mkhatshwa et al 49 considered nonlinear PDEs modeling problems that arise in fluid mechanics. The BSQLM on overlapping grids was introduced by Mkhatshwa et al 38 and implemented to solve time‐dependent Emden‐Fowler equations. This method remains to be generalized and its robustness remains to be tested in the case of highly nonlinear PDEs with strong coupling.…”

“…, +1 and Φ i r , +1 denote values of f θ , , and ϕ at the collocation points. The boundary conditions are imposed on Equations (38) to (40) for each i W = 0, 1, 2, …, . Equations (38) to (40) can be written as a matrix system of the form…”