Open Access Research Article

A Class of Exponentially Fitted Finite Diﬀerence Scheme for Sin-gularly Perturbed Equations Involving Small Delays in Reaction and Convection Terms

Authors

Amar kumar Department of Mathematics, Munger University, Munger, Bihar- 811201, India.
Bipin Prasad Department of Mathematics, Munger University, Munger, Bihar- 811201, India.
Rakesh Ranjan * Government Polytechnic Lakhisarai, Science, Technology and Technical Education Department, Govt. Bihar- 811311, India.

https://doi.org/10.22105/kmisj.v2i4.104

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Abstract

A new exponentially ﬁtted ﬁnite diﬀerence method is introduced for the numerical treatment of a second-order singularly perturbed diﬀerential equation (SPDDE) involving small delays in the ﬁrst derivative and undiﬀerentiated terms. The solution of such equations exhibits left-layer or right-layer behavior in the underlying domain. Taylor’s series expansion procedure is used for constructing an equivalent valid version of the original problem and deriving a new three-term ﬁnite diﬀerence scheme, respectively. The non-uniformity in the solution behavior is resolved by introducing an appropriate exponential ﬁtting parameter in the derived new scheme. The resulting system of equations is solved by the well-known
’discrete invariant algorithm.’ Convergence analysis of the ﬁtted method is discussed, and the theory is illustrated by performing numerical experiments on test example problems. Tabulated computational results show the applicability and accuracy of the method. Theory and computation show that the method is able to approximate the solution very well with a second-order convergence rate. The graphical representation of the solution graph for the tested problems illustrates the impact of varied delay shifts on the layer behavior of the solution.

Keywords

Differential-difference equation Singular perturbation problem Boundary layer Stability and convergence ﬁnite diﬀerence method

Author Biography

Amar kumar, Department of Mathematics, Munger University, Munger, Bihar- 811201, India.

Department of Mathematics, Munger University, Munger, Bihar- 811201, India,

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kumar, A., Prasad, B., & Ranjan, R. (2025). A Class of Exponentially Fitted Finite Diﬀerence Scheme for Sin-gularly Perturbed Equations Involving Small Delays in Reaction and Convection Terms. Karshi Multidisciplinary International Scientific Journal, 2(4), 195-211. https://doi.org/10.22105/kmisj.v2i4.104

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