Numerical treatment of singular perturbation problems exhibiting dual boundary layers

A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method

“…Rai and Sharma [3][4][5] developed some numerical methods for solving singularly perturbed differential-difference equations with turning points. Phaneendra et al [6] employed a fitted operator finite difference method on a uniform mesh for solving singularly perturbed two-point boundary value problems exhibiting dual boundary layers. Becher and Roos [7] proposed a piecewise-uniform numerical method for turning point problems with two exponential boundary layers.…”

“…Such behavior results in some major computational difficulties in the numerical treatment of singularly perturbed problems. Standard finite difference or finite element methods used on uniform meshes can not obtain accurate approximate solutions and in recent years a large number of special-purpose methods have been proposed [1][2][3][4][5][6][7].…”

A Semi-Analytic Method for Solving Singularly Perturbed Twin-Layer Problems With a Turning Point

“…Munyakazi and Patidar developed a fitted-mesh finite difference method with Richardson extrapolation in [28]. For solving Equation (1.1), a fitted-operator scheme was constructed by Phaneendra et al in [34] using nonsymmetric finite differences for the first-order derivatives. Becher and Roos employed a finite difference scheme with Richardson extrapolation for solving the model problem on a piecewise-uniform Shishkin mesh [2].…”

SCEM approach for singularly perturbed linear turning mid-point problems with an interior layer

“…In another great study [2], Parul gave some examples to these kinds of problems occuring in almost all science branches and briefly examined the conventional methods. Various numerical approaches also were applied to approximate to the solutions of turning point problems such as finite difference methods [5], finite element method [6,7], numerical integration method [11], initial value techniques [12] and Reproducing Kernel Method [13,14]. In the detailed work [15], Sharma et al review the theory and methods, and study the development covering the years 1970-2011.…”

“…Some of them can be given as : finite difference methods [5], finite element methods [6,7], the Method of c ⃝ 2016 BISKA Bilisim Technology (WKB) Approximation [8,9], the Method of Matched Asymptotic Expansions (MMAE) [10], numerical integration methods [11], initial value techniques [12] and reproducing kernel methods (RKM) [13,14]. In the detailed work [15], Sharma et al review the theory and methods, and study the development covering the years 1970-2011.…”