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A novel exponentially fitted finite difference method for a class of 2nd order singularly perturbed boundary value problems with a simple turning point exhibiting twin boundary layers

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Abstract

The main aim of this paper is to address a novel exponentially fitted finite difference method for the treatment of a class of 2nd order singularly perturbed boundary value problems in ordinary differential equations with a simple turning point. Solution of such pervasive problem exhibits twin boundary layers when the perturbation parameter \(\varepsilon\) is small tending to zero. The method is most suitable for \(\varepsilon \le 10^{-5}\) and is obtained by partitioning the domain into two subdomains. Taylor’s series with non symmetric difference approximations to the first derivative is used to derive new three term finite difference schemes valid over each of the two subdomains. Non-uniformity in the solution is resolved by the introduction of suitable exponential fitting factors in the derived schemes using the asymptotic theory of singular perturbations. At the turning point, the reduced equation is approximated by the use of central difference analogue of 2nd order derivative. Thomas algorithm is implemented on \(Code::Blocks\, IDE\, for\, Fortran-90\) platform for solving the resulting tridiagonal system of equations. Stability and Convergence of the method are analysed. Efficiency of the method is illustrated by solving three standard problems for \(\varepsilon \le 10^{-5}\) and presenting the results in tabular/graphical form. \(A\, new\, formula\) is introduced and used to know how much a method overcomes the other method(s). Comparisons made show the capability of the method in producing highly accurate and uniformly convergent results with linear rate for all the values of the mesh size \(h>>\varepsilon\).

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Abbreviations

\(\varepsilon\) :

Singular perturbation parameter

\(z\,\text {and}\,t\) :

Dependent and Independent variable respectively

h :

Step length

C :

Generic positive constant independent of \(\varepsilon\) and h

\(\sigma (\rho )\) :

Constant fitting factor

\(\Omega\) and \({\bar{\Omega }}\) :

Open and close interval respectively

\(\omega\) :

Mesh function

N :

An integer denoting the number of subinterval

\(\Phi \, \text {and}\,\gamma\) :

Finite constant

\(\alpha (t),\,\beta (t)\) and r(t):

Sufficiently smooth functions

z(t):

Solution of continuous problem

\(z_i\) :

Numerical solution

\(\tau _i\) :

Truncation error

\(E_{\varepsilon }^{N},\,\)E\(_{\varepsilon }^{2N}\) :

Maximum absolute errors

\(R_{\varepsilon }^{N},\, R_{p\varepsilon }^{N}\) :

Rate of convergence

\({\pi ^ \pm }\) :

Barrier function

\(L_{\tau }\) :

Linear differential operator

\(O(.),\,o(.)\) :

Landau order symbols

\(\,{\left\| . \right\| _{h,\infty }}\) :

Discrete \({l_\infty }\) -norm

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  1. Department of Mathematics, National Institute of Technology Jamshedpur, Adityapur, Jamshedpur, Jharkhand, 831014, India

    Rakesh Ranjan & Hari Shankar Prasad

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  1. Rakesh Ranjan
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  2. Hari Shankar Prasad
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Correspondence to Rakesh Ranjan.

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Ranjan, R., Prasad, H.S. A novel exponentially fitted finite difference method for a class of 2nd order singularly perturbed boundary value problems with a simple turning point exhibiting twin boundary layers. J Ambient Intell Human Comput 13, 4207–4221 (2022). https://doi.org/10.1007/s12652-022-03902-0

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