A Special Case of Variational Formulation for Two-Point Boundary Value Problem in L2(Ω)

Pedro Pablo Cárdenas Alzate

Department of Mathematics, Universidad Tecnológica de Pereira, Pereira, Colombia.

DOI: 10.4236/am.2015.64065                

Abstract

We consider the nonlinear boundary value problems for elliptic partial differential equations and using a maximum principle for this problem we show uniqueness and continuous dependence on data. We use the strong version of the maximum principle to prove that all solutions of two-point BVP are positives and we also show a numerical example by applying finite difference method for a two-point BVP in one dimension based on discrete version of the maximum principle.

Keywords

Two-Point Boundary, Variational Problem, Stability Restriction

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Larsson S. and Thomée, V. (2009) Partial Differential Equations with Numerical Methods. Springer-Verlag, New York.
[2]	McRea, G.J. and Godin, W.R. (1967) Numerical Solution of Atmospheric Diffusion for Chemically Reacting Flows. Journal of Computational Physics, 77, 1-42.
[3]	Cárdenas Alzate, P.P. (2014) A Survey of the Implementation of Numerical Schemes for Linear Advection Equation. Advances in Pure Mathematics, 4, 467-479. http://dx.doi.org/10.4236/apm.2014.48052
[4]	Cárdenas Alzate, P.P. (2014) A Survey of the Implementation of Numerical Schemes for the Heat Equation Using Forward Euler in Time. Journal of Applied Mathematics and Physics, 2, 1153-1158. http://dx.doi.org/10.4236/jamp.2014.213135
[5]	Hundsdorfer, W. and Koren, B. (1995) A Positive Finite-Difference Advection Scheme Applied on Locally Refined Grids. Journal of Computational Physics, 117, 35-36. http://dx.doi.org/10.1006/jcph.1995.1042
[6]	Canuto, C. and Hussaini, M. (1988) Spectral Methods in Fluids Dynamics, Springer Series in Computational Physics. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-642-84108-8
[7]	Dehghan, M. (2007) The One-Dimensional Heat Equation Subject to a Boundary Integral Specification. Chaos, Solitons & Fractals, 32, 661-675. http://dx.doi.org/10.1155/MPE.2005.61
[8]	Lu, X, Tervola, P. and Viljanen, M. (2005) A New Analytical Method to Solve the Heat Equation for a Multi-Dimen-sional Composite Slab. Journal of Physics, 38, 2873. http://doi:10.1088/0305-4470/38/13/004
[9]	Eriksson, K., Estep, D., Hansbo, P. and Johnson, C. (1996) Computational Differential Equations. Cambridge University Press, Cambridge.