Palabras clave
multiresolution analysis
wavelet-Galerkin
B-spline
análisis multirresolución
wavelet-Galerkin
ondeletas Galerkin
Cómo citar
Resumen
En este trabajo se desarrolla un método Wavelet-Galerkin Adaptativo para la resolución de ecuaciones diferenciales parabólicas que modelan problemas físicos, con diferentes escalas en el espacio y en el tiempo. Se utiliza un esquema semi-implícito en diferencias temporales y la estructura multirresolución de las B-splines sobre intervalo.Como es sabido que en muchos casos las soluciones presentan gradientes localmente altos, se han diseñado estimadores locales de error y una estrategia adaptativa eficiente para elegir la escala apropiada en cada tiempo. Finalmente, se realizaron experimentos que ilustran la aplicabilidad y la eficiencia del método propuesto.
Citas
Bertoluzza, S.; Naldi, G. (1996) “A wavelet collocation method for the numerical solution of partial differential equations”, Applied and Computational Harmonic Analysis 3(1): 1–9.
Bindal, A.; Khinast, J.G.; Ierapetritou, M.G. (2003) “Adaptive multi- scale solution of dynamical systems in chemical processes using wavelets”, Computers and Chemical Engineering 27(1): 131–142.
Burgers, J.M. (1948) “A mathematical model illustrating the theory of turbulence”, Adv. Appl. Mech. 1: 171–199.
Cammilleri, A; Serrano, E.P. (2001) “Spline multiresolution analysis on the interval”, Latin American Applied Research 31(2): 65–71.
Chui, C.K. (1992) An Introduction to Wavelets. Academic Press, New York.
Ciarlet, P.G. (1978) The Finite Element Method for Elliptic Problems. North Holland, New York.
Kumar, B V.; Mehra, M. (2005) “Wavelet-Taylor Galerkin method for the Burgers equation”, BIT Numerical Mathematics 45: 543–560.
Kumar, V.; Mehra, M. (2007) “Cubic spline adaptive wavelet scheme to solve singularly perturbed reaction diffusion problems”, International Journal of Wavelets, Multiresolution and Information Processing 5: 317–
Lin, E.B.; Zhou, X. (2001) “Connection coefficients on an interval and wavelet solutions of Burgers equation”, Journal of Computational and Applied Mathematics 135(1): 63–78.
Mallat, S.G. (2009) A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press - Elsevier, MA EE.UU.
Quraishi, S.M.; Gupta, R.; Sandeep, K. (2009) “Adaptive wavelet Galerkin solution of some elastostatics problems on irregularly spaced nodes”, The Open Numerical Methods Journal 1: 20–26.
Schöenberg, I.J. (1969) “Cardinal interpolation and spline functions”, Jour- nal of Approximation Theory 2: 167–206.
Schult, T.L.; Wyld, H.W. (1992) “Using wavelets to solve the Burgers equation: A comparative study”, Physical Review A 46(12): 7953–7958.
Vampa, V.; Martín, M.T.; Serrano, E. (2010) “A hybrid method using wavelets for the numerical solution of boundary value problems on the interval”, Appl. Math. Comput. 217(7): 3355–3367.
Vampa, V. (2011) Desarrollo de Herramientas Basadas en la Transformada Wavelet para su Aplicación en la Resolución Numérica de Ecuaciones Diferenciales. Tesis de Doctorado en Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina.
Vampa, V.; Martín, M.T.; Serrano, E. (2012) “A new refinement Wavelet- Galerkin method in a spline local multiresolution analysis scheme for boundary value problems”, Int. Journal of Wavelets, Multiresolution and Information Processing 11(2), 19 pp.
Vasilyev, O.V.; Paolucci, S. (1996) “A dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain”, Journal of Computational Physics 125(2): 498–512.
Walnut, D.F. (2002) An Introduction to Wavelet Analysis. Applied and Nu- merical Harmonic Analysis Series, Birkhäuser, Boston.
Whitham, G.B. (1974) Linear and Nonlinear Waves, Wiley, New York.