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Abstract
A numerical method for solving a class of quasi-linear singular two-point boundary value problems with a transition layer is presented in this paper. For the problem ϵ u xx +a(u+f(x))u x +b(x, u)=0, we develop a multiple scales method. First, this method solves the location of the transition layer, then it approximates the singular problem with reduced problems in the non-layer domain and pluses a layer corrected problem which nearly has an effect in the layer domain. Both problems are transformed into first-order problems which can be solved easily. For the problem ϵ u xx +b(x, u)=0, we establish a similar method which approximate the problem with reduced problems and a two-point boundary value problem. Unsteady problems are also considered in our paper. We extend our method to solve Burgers’ equation problems by catching the transition layer with the formula of shock wave velocity and approximating it by a similar process.
Acknowledgements
The support from the National Natural Science Foundation of China (No. 10671146, No. 50678122) is fully acknowledged. The authors are grateful to the Prof. Mingkang Ni for his valuable discussion on Examples 1, 2 and 4. The authors are deeply grateful and sincerely obliged to the referees for their valuable comments and enlightening suggestions which has led to a great improvement of this paper.