Research ArticleOpen AccessGoogle Scholar indexed
Parabolic Partial Differential Equations with Border Conditions of Dirichlet as Inverse Moments Problem
Grupo de Aplicaciones Matematicas y Estadisticas de la Facultad de Ingenieria (GAMEFI), Universidad Nacional de La Plata, Buenos Aires, Argentina
Departamento de Matematica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina
- 1 Grupo de Aplicaciones Matematicas y Estadisticas de la Facultad de Ingenieria (GAMEFI), Universidad Nacional de La Plata, Buenos Aires, Argentina
- 2 Departamento de Matematica, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Buenos Aires, Argentina
Copy link · social · email
Abstract
We considerer parabolic partial differential equations: under the conditions , on a region . We will see that an approximate solution can be found using the techniques of generalized inverse moments problem and also bounds for the error of estimated solution. First we transform the parabolic partial differential equation to the integral equation . Using the inverse moments problem techniques we obtain an approximate solution of . Then we find a numerical approximation of when solving the integral equation , because solving the previous integral equation is equivalent to solving the equation .
KeywordsParabolic PDEsIntegral EquationsGeneralized Moment Problem
- Pintarelli, M.B. and Vericat, F. (2014) Partial Differential Equations as Three-Dimensional Inverse Problem of Moments. Journal of Mathematics and System Science, 4, 657-666.
- Pintarelli, M.B. (2016) Parabolic Partial Differential Equations as Inverse Moments Problem. Applied Mathematics, 7, 77-99. https://doi.org/10.4236/am.2016.71007
- Cherniha, R.M. (2001) New Exact Solutions of One Nonlinear Equation in Mathematical Biology and Their Properties. Ukrainian Mathematical Journal, 53, 393-411. https://doi.org/10.1023/A:1015252112419
- Mittal, R.C. and Jiwari, R. (2011) A Higher Order Numerical Scheme for Some Nonlinear Differential Equations: Models in Biology. International Journal for Computational Methods in Engineering Science and Mechanics, 12, 134-140. https://doi.org/10.1080/15502287.2011.564265
- Forsythe, G.E. and Wasow, W.R. (1960) Finite Difference Methods for Partial Differential Equations. John Wiley and Sons, New York.
- Zafarullah, A. (1971) Some Stable Implicit Difference Methods for Heat Equation with Derivative Boundary Condition. The Computer Journal, 14, 309-311. https://doi.org/10.1093/comjnl/14.3.309
- Keast, P. and Mitchell, A.R. (1966) On the Instability of the Crank Nicholson Formula under Derivative Boundary Conditions. The Computer Journal, 9, 110-114. https://doi.org/10.1093/comjnl/9.1.110
- Ames, W.F. (1992) Numerical Methods for Partial Differential Equations. Academic Press, Inc., New York.
- Lapidus, L. and Pinder, G.F. (2011) Numerical Solution of Partial Differential Equations in Science and Engineering. John Wiley and Sons, New York.
- Friedman, A. (2013) Partial Differential Equations of Parabolic Type, Courier Corporation.
- Knabner, P. and Angerman, L. (2006) Numerical Methods for Elliptic and Parabolic Partial Differential Equations. Springer, Berlin.
- Johnson, C. (2009) Numerical Solution of Partial Differential Equations by the Finite Element Meth. Dover Publications, Inc., New York.
- Nayroles, B., Touzot, G. and Villon, P. (1992) Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements. Computational Mechanics, 10, 307-318. https://doi.org/10.1007/BF00364252
- Babuska, I. and Melenk, J. (1997) The Partition of Unity Method. International Journal for Numerical Methods in Engineering, 40, 727-758. https://doi.org/10.1002/(SICI)1097-0207(19970228)40:4 3.0.CO;2-N