Volume 8, no. 4Pages 5 - 13

# An Integral Method for the Numerical Solution of Nonlinear Singular Boundary Value Problems

M.V. Bulatov, P.M. Lima, Thanh Do Tien
We discuss the numerical treatment of a nonlinear singular second order boundary value problem in ordinary differential equations, posed on an unbounded domain, which represents the density profile equation for the description of the formation of microscopic bubbles in a non-homogeneous fluid. Due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational effort. This is the motivation for our present study, where we describe an alternative approach, based on the reduction of the original problem to an integro-differential equation. In this way, we obtain a Volterra integro-differential equation with a singular kernel. The numerical integration of such equations is not straightforward, due to the singularity. However, in this paper we show that this equation may be accurately solved by simple product integration methods, such as the implicit Euler method and a second order method, based on the trapezoidal rule. We illustrate the proposed methods with some numerical examples.
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Keywords
density profile equation; singular boundary value problem; integro-differential equation; implicit Euler method.
References
1. Gurtin M.E., Polignone D., Vinals J. Two-Phase Binary Fluids and Immiscible Fluids Described by an Order Parameter. Mathematical Models and Methods in Applied Sciences, 1996, vol. 6, pp. 815-831. DOI: 10.1142/S0218202596000341
2. Dell'Isola F., Gouin H., Rotoli G. Nucleation of Spherical Shell-Like Interfaces by Second Gradient Theory: Numerical Simulations. European Journal of Mechanics - B/Fluids, 1996, vol. 15, pp. 545-568.
3. Gavrilyuk S.L., Shugrin S.M. Media with Equations of State that Depend on Derivatives. Journal of Applied Mechanics and Technical Physics, 1996, vol. 37, pp. 177-189. DOI: 10.1007/BF02382423
4. Lima P.M., Chemetov N.V., Konyukhova N.B., Sukov A.I. Analytical-Numerical Investigation of Bubble-Type Solutions of Nonlinear Singular Problems. Journal of Computational and Applied Mathematics, 2006, vol. 189, pp. 260-273. DOI: 10.1016/j.cam.2005.05.004
5. Kitzhofer G., Koch O., Lima P.M., Weinmuller E. Efficient Numerical Solution of the Density Profile Equation in Hydrodynamics. Journal of Scientific Computing, 2007, vol. 32, pp. 411-424. DOI: 10.1007/s10915-007-9141-0
6. Konyukhova N.B., Lima P.M., Morgado M.L., Soloviev M.B. Bubbles and Droplets in Nonlinear Physics Models: Analysis and Numerical Simulation of Singular Nonlinear Boundary Value Problems. Computational Mathematics and Mathematical Physics, 2008, vol. 48, no. 11, pp. 2018-2058. DOI: 10.1134/S0965542508110109
7. Derrick G. Comments on Nonlinear Wave Equations as Models for Elementary Particles. Journal of Mathematical Physics, 1965, vol. 5, pp. 1252-1254. DOI: 10.1063/1.1704233
8. Gazzola F., Serrin J., Tang M. Existence of Ground States and Free Boundary Problems for Quasilinear Elliptic Operators. Advances in Differential Equations, 2000, vol. 5, pp. 1-30.
9. Hastermann G., Lima P.M., Morgado M.L., Weinmuller E.B. Density Profile Equation with p-Laplacian: Analysis and Numerical Simulation. Applied Mathematics and Computation, 2013, vol. 225, pp. 550-561. DOI: 10.1016/j.amc.2013.09.066
10. Kulikov G.Yu., Lima P.M., Morgado M.L. Analysis and Numerical Approximation of Singular Boundary Value Problems with the p-Laplacian in Fluid Mechanis. Journal of Computational and Applied Mathematics, 2014, vol. 262, pp. 87-104. DOI: 10.1016/j.cam.2013.09.071
11. Weiss R., Anderssen R.S. A Product Integration Method for a Class of Singular First Kind Volterra Equations. Numerische Mathematik, 1972, vol. 18, pp. 442-456. DOI: 10.1007/BF01406681
12. Weiss R. Product Integration for the Generalized Abel Equations. Mathematics of Computation, 1972, vol. 26, pp. 177-186. DOI: 10.1090/S0025-5718-1972-0299001-7
13. Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge, University Press, 2004. DOI: 10.1017/CBO9780511543234
14. Brunner H., Van Der Houven P.J. The Numerical Solution of Volterra Equations. Amsterdam, North-Holland, CWI Monographs 3, 1986.
15. Linz P. Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia, 1985. DOI: 10.1137/1.9781611970852
16. Brunner H. 1896-1996: One Hundred Years of Volterra Integral Equations of the First Kind. Applied Numerical Mathematics, 1997, vol. 24, pp. 83-93. DOI: 10.1016/S0168-9274(97)00013-5