Wavelet method for solving second-order quasilinear parabolic equations with a conservative principal part. (Russian, English) Zbl 1199.65328

Abbasov, E. M.; Dyshin, O. A.; Suleimanov, B. A.

Zh. Vychisl. Mat. Mat. Fiz. 49, No. 9, 1629-1642 (2009); translation in Comput. Math., Math. Phys. 49, No. 9, 1554-1566 (2009).

Summary: A method based on wavelet transforms is proposed for finding classical solutions to initial-boundary value problems for second-order quasilinear parabolic equations. For smooth data, the convergence of the method is proved and the convergence rate of an approximate weak solution to a classical one is estimated in the space of wavelet coefficients. An approximate weak solution of the problem is found by solving a nonlinear system of equations with the help of gradient-type iterative methods with projection onto a fixed subspace of basis wavelet functions.

MSC:

65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K20	Initial-boundary value problems for second-order parabolic equations
35K57	Reaction-diffusion equations
65T60	Numerical methods for wavelets
35D30	Weak solutions to PDEs

Keywords:

weak and approximate weak solutions to initial-boundary value problems for parabolic equations; multiresolution analysis; wavelet basis; gradient-type iterative method; irregular operator equation