A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs
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A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs Pramod Chakravarthy Podila1 · Kamalesh Kumar1 Received: 15 November 2019 / Revised: 12 March 2020 / Accepted: 16 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this study, we consider the time-delayed singularly perturbed parabolic PDEs (SPPPDEs). We know that the classical finite difference scheme will not produce good results for singular perturbation problems on a uniform mesh. Here, we propose a new stable finite difference (NSFD) scheme, which produces good results on a uniform mesh and also on an adaptive mesh. The NSFD scheme is constructed based on the stability of the analytical solution. Results are compared with the results available in the literature and observed that the proposed method is efficient over the existing methods for solving SPPPDEs. Keywords Singular perturbation · Delay partial differential equations · Stable finite difference scheme Mathematics Subject Classification 65M06 · 65L11
1 Introduction In recent years, the study of singularly perturbed delay differential equations is attracting many researchers because of its applications in diverse fields such as biosciences, control theory, economics, material science, tumor growth, neural networks, and robotics etc., (Wu 1996; Musila and Lansky 1991; Arino et al. 2006; Murray 2001; Cheng and Jia-qi 2005). Many physical phenomena that display time-delayed or memory effect can be modeled by PDEs with a delay term (Arino et al. 2006). Lange and Miura (1982), developed asymptotic analysis of differential-difference equations in ODEs with perturbations. Later, several researchers started working on numerical
Communicated by Frederic Valentin.
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Pramod Chakravarthy Podila [email protected] Kamalesh Kumar [email protected]
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Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India 0123456789().: V,-vol
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solutions of singularly perturbed differential difference equations. We refer Roos et al. (2008) for various numerical methods for singular perturbation problems (SPPs). Ansari and Shishkin (2007) proposed a numerical method for SPPPDEs with a delay using central finite difference operators on Shishkin mesh. Gowrisankar and Natesan (2017), developed a scheme for SPPPDEs on Shishkin type meshes. They had given the error estimates and the stability analysis of their method. The Richardson extrapolation technique has been used in Das and Natesan (2018), to enhance the order of convergence. Das and Natesan (2015) developed a hybrid numerical scheme, which is a combination of the central difference scheme and midpoint upwind scheme for the numerical approximation of time-delayed SPPPDEs. To solve SPPPDEs, Salama and AI-Amery (2017) discretized the time derivative using the Crank–Nicolson scheme and proposed a fourth-order compact scheme to solve the resulting system of ordinary differen
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