Numerical treatment of singularly perturbed two point boundary value problems |
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| Institution: | 1. Department of Chemical Engineering, Konkuk University, Seoul 05029, Republic of Korea;2. Intelligent Construction Automation Center, Kyungpook National University, 80, Daehak-ro, Buk-gu, Daegu, 41566, Republic of Korea;1. AGH University of Science and Technology, Al. Mickiewicza 30, 30-059, Kraków, Poland;2. Department of Systems and Computer Networks, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland;1. Swanson Hip and Knee Center of Excellence and Research Institute Desert Orthopaedic Center, Las Vegas, NV, USA;2. Desert Orthopedic Center & Swanson Hip/Knee Research Foundation, 2800 E.Desert Inn, Suit 100, Las Vegas, NV, 89121, USA;1. Department of Applied Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi 110021, India;2. Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA 71272, USA;1. Supercomputer Education Research Center, Indian Institute of science, Bangalore 560012, India;2. Department of Mathematics, Indian Institute of Technology, Guwahati 781039, India |
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| Abstract: | We propose a method for numerically solving linear singularly perturbed two point boundary value problems in ordinary differential equations with a boundary layer on the left end of the underlying interval. This is a practical method and can be easily implemented on a computer. The original problem is divided into inner and outer region differential equation systems. The reduced problem is solved to obtain the terminal boundary condition. Then, a new inner region problem is created and solved as a two point boundary value problem (TPBVP). In turn, the outer region problem is also solved as a TPBVP. Both these TPBVPs are efficiently treated by employing a slightly modified classical finite difference scheme coupled with discrete invariant imbedding algorithm to obtain the numerical solutions. The stability of some recurrence relations involved in the algorithm is investigated. The proposed method is iterative on the terminal point. Some numerical examples are included, and the computational results are compared with exact solutions. It is observed that the accuracy predicted can always be achieved with very little computational effort. |
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