Numerical solution of doubly singular boundary value problems by finite difference method
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Numerical solution of doubly singular boundary value problems by finite difference method Pradip Roul1
· V. M. K. Prasad Goura1
Received: 25 November 2019 / Revised: 7 September 2020 / Accepted: 28 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we propose a computational technique based on a combination of optimal homotopy analysis method (OHAM) and an iterative finite difference method (FDM) for a class of derivative-dependent doubly singular boundary value problems: ( p(x)y ) = q(x) f (x, y(x), y (x)), 0 ≤ x ≤ 1, y (0) = 0, α y(1) + β y (1) = B or
y(0) = A, α y(1) + β y (1) = B.
The principal idea of this approach is to decompose the domain of the problem D = [0, 1] into two subdomains as D = D1 ∪ D2 = [0, γ ] ∪ [γ , 1] (γ is the vicinity of the singularity). In the first domain D1 , we use OHAM to overcome the singularity behaviour at x = 0. In the second domain D2 , a FDM is designed for solving the resulting regular boundary value problem. Convergence analysis of the method is carried out. Three nonlinear examples are considered to demonstrate the performance and accuracy of the proposed method. It is shown that the computational order of convergence of the FDM is two. Keywords Derivative-dependent source function · Singular boundary value problems · OHAM · FDM · Convergence analysis Mathematics Subject Classification 65L10 · 65L12
Communicated by Yimin Wei.
B
Pradip Roul [email protected] V. M. K. Prasad Goura [email protected]
1
Department of Mathematics, Visvesvaraya National Institute of Technology, Nagpur 440010, India 0123456789().: V,-vol
123
302
Page 2 of 25
P. Roul, V. M. K. P. Goura
1 Introduction We consider a class of derivative-dependent doubly singular boundary value problems (DDDSBVP) of the form: ( p(x)y (x)) = q(x) f (x, y(x), y (x)), 0 ≤ x ≤ 1,
(1)
with the boundary conditions (BCs):
or
y (0) = 0, α y(1) + β y (1) = B
(2)
y(0) = A, α y(1) + β y (1) = B,
(3)
where A, α ≥ 0, β > 0 and B are finite constants. We assume that (A-1) i. p(x) = x b0 g(x), 0 ≤ b0 < 1, g(0) = 0, and p(0) = 0 and p(x) > 0 in (0,1]. 1 1 dη < ∞. ii. p(x) ∈ C[0, 1] C 1 (0, 1] and 0 p(η) iii. S(x) ∈ C 2 (0, 1], where S(x) =
1 g(x)
on (0,1].
(A-2) i. q(x) > 0 in (0,1], q(x) is unbounded near x = 0 and
1 0
q(x)dx < ∞.
(A-3) i. The function f (x, y, z) is continuous, where z = y , ii. ∂∂ yf and ∂∂zf exists and are continuous, iii.
∂f ∂y
≥ 0 and | ∂∂zf | ≤ L for some positive constant L.
Since p(0) = 0 and q(x) is discontinuous at x = 0, the problem (1) is doubly singular (Bobisud 1990). Existence and uniqueness of the solutions to the singular boundary value problems (SBVPs) with different types of BCs have been discussed in Bobisud (1990), Dunninger and Kurtz (1986), Pandey and Verma (2010), Zhang (1996) and Zhang (1993). For example, Bobisud (1990) established existence results for the differential equation ( p(x)y (x)) = q(x) f (x, y(x), p(x)y (x)), 0 ≤ x ≤ 1, with the followi
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