Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through t
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Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through tension spline P. Murali Mohan Kumar1,2
· A. S. V. Ravi Kanth2
Received: 7 June 2019 / Revised: 11 July 2020 / Accepted: 26 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This article contributes a numerical technique for a class of singularly perturbed time delayed parabolic partial differential equation. A priori results of maximum principle, stability and bounds are discussed. The continuous problem is semi-discretized by the Crank–Nicolson based scheme in the temporal direction and then discretized by the tension spline scheme on non-uniform Shishkin mesh. Error estimation for the discretized problem is derived. To validate the theoretical findings, the numerical outcomes for linear and nonlinear problems are tested. Keywords Nonlinear differential-difference equation · Tension spline · Shishkin mesh · Singular perturbation problems Mathematics Subject Classification 65M12 · 65L11 · 35K20
1 Introduction Consider a class of singularly perturbed time delayed parabolic partial differential equation on (t, x) ∈ (0, T ] × Ω: (∂t + £ε ) u(t, x) = f (t, x) − b(t, x)u(t − τ, x),
Communicated by Frederic Valentin.
B
A. S. V. Ravi Kanth [email protected] P. Murali Mohan Kumar [email protected]; [email protected]
1
Department of Basic Sciences and Humanities, GMR Institute of Technology, Rajam, Srikakulam, Andhra Pradesh 532 127, India
2
Department of Mathematics, National Institute of Technology Kurukshetra, Kurukshetra, Haryana 136 119, India 0123456789().: V,-vol
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P. M. M. Kumar, A. S. V. Ravi Kanth
subject to initial condition u(t, x) = B (t, x), (t, x) ∈ Λ B = [−τ, 0] × [0, 1],
(2)
and boundary conditions u(t, 0) = L (t), on Λ L = {(t, 0) : 0 ≤ t ≤ T },
(3)
u(t, 1) = R (t), on Λ R = {(t, 1) : 0 ≤ t ≤ T },
(4)
where £ε = −ε
∂ 2 u(t, x) + a(t, x)u(t, x). ∂x2
Here τ > 0 is the delay parameter, and 0 < ε ≤ 1 is the singular perturbation parameter. Assume f (t, x), b(t, x), a(t, x), L (t, x), R (t, x) and B (t, x) are sufficiently smooth and bounded functions that satisfy a(t, x) ≥ 0, b(t, x) ≥ γ > 0, (t, x) ∈ [0, T ] × [0, 1]. Also, assume that T is the terminal time which satisfy the condition T = κτ , for κ > 0. The initial function (t, x) is assumed to satisfy the compatibility conditions (Protter and Weinberger 1967): B (0, 0) = L (0),
(5)
B (0, 1) = R (0),
(6)
∂ L (0) ∂ 2 B (0, 0) + a(0, 0) B (0, 0) = f (0, 0) − b(0, 0) B (−τ, 0), −ε ∂t ∂x2
(7)
∂ R (0) ∂ 2 B (0, 1) + a(0, 1) B (0, 1) = f (0, 1) − b(0, 1) B (−τ, 1). −ε ∂t ∂x2
(8)
and
The problem (1)–(4) has a unique solution (Ansari et al. 2007) with the above conditions and assumptions. Singularly perturbed delay differential equations are in to the picture in different fields of science and engineering, for instance, control theory (Van Harten and Schumacher 1978), chemical processes (Adomian and R
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