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Solving frontier problems modelled by nonlinear partial differential equations. (English) Zbl 0767.35016

Summary: We present a further development of the decomposition method [the author, ibid. 21, No. 5, 101-127 (1991; Zbl 0732.35003), and Nonlinear stochastic operator equations (1986; Zbl 0609.60072)], which leads to a single efficient and global method of solving linear or nonlinear, ordinary or partial differential equations for initial-value or boundary-value problems. No linearization, perturbation, or resort to discretized methods is involved. Potential savings in computation are very large (perhaps six orders of magnitude in some cases) and important implications exist for modeling and computational analysis. It is to be noted that once we realize that we can be less constrained by the mathematics by removing the necessity of techniques such as linearization, perturbation, or discretization to make analysis and computation of the models feasible and practical; we become able to develop more sophisticated and realistic models. Modeling effectively means retention of essential features while striving for simplicity so that the resulting equations can be solved. With fewer limitations imposed to achieve tractability, the models can be more realistic and we have a convenient and global technique for solution.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
35G30 Boundary value problems for nonlinear higher-order PDEs
35A35 Theoretical approximation in context of PDEs
35R60 PDEs with randomness, stochastic partial differential equations
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References:

[1] Adomian, G., Nonlinear Stochastic Operator Equations (1986), Academic Press · Zbl 0614.35013
[2] Adomian, G., A review of the decomposition method and some recent results for nonlinear equations, Int. J. Comput. and Math. with Applic., 21, 5, 101-127 (1991) · Zbl 0732.35003
[3] Adomian, G.; Rach, R., Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Int. J. Comput. and Math. with Applic., 19, 12, 9-12 (1990) · Zbl 0702.35058
[4] Cherruault, Y., Convergence of Adomian’s method, Kybernete, 18, 2, 31-38 (1989) · Zbl 0697.65051
[5] Adomian, G., Decomposition solution of nonlinear hyperbolic equations, (Working papers presented at the Proc. Seventh Int. Conf. on Math. Modeling. Working papers presented at the Proc. Seventh Int. Conf. on Math. Modeling, Chicago (1989)) · Zbl 0729.65072
[6] Adomian, G., Nonlinear Stochastic System Theory and Applications to Physics (1989), Kluwer · Zbl 0698.35099
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