Abstract
A new canonical formalism for solving general nonlinear systems is presented. Fundamental to this formalism is the construction of forward (advanced) and backward (retarded) propagators that yield the problem's solution exactly, by propagating volume, surface, and initial sources. These propagators also satisfy reciprocity and semigroup properties. Therefore, they represent a generalization to nonlinear systems of the Green's functions from linear theory. Several examples are presented to illustrate the application of the algorithm as well as its numerical advantages over alternative methods.
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References
Cacuci, D. G., Perez, R. B., Protopopescu, V., “Duals and Propagators: A Canonical Formalism for Nonlinear Equations,” J. Math. Phys., 29, 353–361 (1988); see also D. G. Cacuci, R. B. Perez, V. Protopopescu, “Extending Green's Function Formalism to General Nonlinear Equations,” in Proceedings of the IVth Workshop on Nonlinear Evolution Equations and Dynamical Systems, ed. J.J.P. Léon, World Scientific, Singapore, 1988, pp. 87–95.
Cacuci, D. G., Protopopescu, V., “Propagators for Nonlinear Systems,” J. Phys. A., (1989), 22 (1989).
Coddington, E. A., Levinson, N., Theory of Ordinary Differential Equations, McGraw Hill, New York, 1955.
Azmy, Y. Y., Cacuci, D. G., Protopopescu, V., “A Comparison Between the Methods of Decomposition and Propagators for Nonlinear Equations,” unpublished report.
Cacuci, D. G., Karakashian, O., “A New Method for Solving the Generalized Burgers-Korteweg-de Vries Equation,” J. Comp. Phys., submitted.
Zabuski, N. J., Kruskal, M. D., “Interactions of “Solitons” in a Collisionless Plasma and the Recurrence of Initial States,” Phys. Rev. Lett., 15, 240–243 (1965).
Taha, T. R., Ablowitz, M. J., “Analytical and Numerical Aspects of Certain Nonlinear Evolution Equations. III Numerical Korteweg-de Vries Equation,” J. Comp. Phys., 55, 231–253 (1984).
Dougalis, V. A., Karakashian, O., “On Some High-Order Accurate Fully Discrete Galerkin Methods for the Korteweg-de Vries Equation,” Math. Comp., 45, 329–345 (1985).
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© 1991 Springer-Verlag
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Cacuci, D.G., Protopopescu, V. (1991). Canonical propagators for nonlinear systems: Theory and sample applications. In: Toscani, G., Boffi, V., Rionero, S. (eds) Mathematical Aspects of Fluid and Plasma Dynamics. Lecture Notes in Mathematics, vol 1460. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091360
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DOI: https://doi.org/10.1007/BFb0091360
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