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An iterative technique for solving a class of local and nonlocal elliptic boundary value problems Randhir Singh1  Received: 7 March 2019 / Accepted: 2 July 2020 © Springer Nature Switzerland AG 2020

Abstract An optimal iterative method is proposed for a reliable solution of a class of Bratutype, Troesch’s and nonlocal elliptic boundary value problems (BVPs). Due to the presence of parameter 𝛿 as well as strong nonlinearity, these problems pose difficulties in obtaining their solutions. With the help of Green’s function theory, we first transform the BVP into an equivalent integral equation, followed by applying the optimal homotopy analysis method to get the approximate solution of high accuracy level. Several examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The numerical results confirm the reliability of the present method as it tackles such nonlocal problems without any limiting assumptions. The convergence and error analysis of the proposed method is discussed. Keywords  Iterative method · Chemical reaction–diffusion · Bratu-type problems · Troesch’s problems · Heat transfer processes

1 Introduction Nonlinear BVPs of ordinary differential equations play an important role in many fields of science and engineering. Two-point (local) and nonlocal BVPs occurs in a wide variety of problems, including the modeling of chemical reactions, and heat transfer [1–5]. We consider a highly nonlinear BVPs, the Bratu’s problem is defined by { �� u + 𝛿eu = 0, x ∈ (0, 1), (1.1) u(0) = u(1) = 0.

* Randhir Singh [email protected] 1



Department of Mathematics, Birla Institute of Technology Mesra, Ranchi 835215, India

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Journal of Mathematical Chemistry

The Bratu’s problem is a highly nonlinear BVP and extensively applied as a benchmark problem to test the accuracy of many analytical and numerical techniques. In the past two decades, several methods have been developed and compared to the exact solution of the Bratu’s problem including Adomian decomposition method (ADM) [6, 7], variational iteration method (VIM) [8, 9], B-spline method [10], non-polynomial spline method [11], Sinc–Galerkin method [12], Lie-group method [13], shooting method [14], neural network method [15], compact finite difference method [16], fourth-order B-spline collocation method [17], and optimal HAM [18]. We next consider a BVP, Troesch’s problem, defined by { �� u − 𝛿 sinh 𝛿u = 0, x ∈ (0, 1), (1.2) u(0) = 0, u(1) = 1, where 𝛿 is a positive constant. The Troesch’s problem arises in an investigation of the confinement of a plasma column by radiation pressure [19] and also in the theory of gas porous electrodes [20, 21]. The closed form solution of this problem [22] in term of the Jacobian elliptic function sc(𝛿|m) is ) ( � u (0) 2 sc(𝛿x|m) u = sinh−1 𝛿 2 �

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where m = 1 − u (0) and satisfies sc(𝛿|m)(1 − m)1∕2 = sinh( 𝛿2 )  . This problem is 4 inherently unstable and difficult, especially when the sensitivity parameter 𝛿 is large. Therefore, Troesch’s problem has bec