FOURIER METHOD FOR A QUASILINEAR PARABOLIC EQUATION WITH PERIODIC BOUNDARY CONDITION

Abstract

FOURIER METHOD FOR A QUASILINEAR PARABOLIC EQUATION WITH PERIODIC BOUNDARY CONDITION

Abstract

A multidimensional mixed problem with Neuman type periodic boundary condition is studied for the quasilinear parabolic equation ∂u
∂t −
a
2 ∂
2u
∂x2 = f(t, x, u). The existence, uniqueness and also continuity of
the weak generalized solution is proved.

Keywords

• Quasilinear parabolic equation,Mixed problem,Fourier method,Periodic boundary condition,Generalized solutions

References

1. Amiraliyev, G. M. and Mamedov, Y. D. Difference scheme on the uniform mesh for singular perturbed pseudoparabolic equations. Turk. J. of Mathematics 19, 207–222, 1995.
2. Chandirov, G. I. On mixed problem for a class of quasilinear hyperbolic equation (PhD. Thesis, Tibilisi, 1970).
3. Colton, D. The exterior Dirichlet problem for ∆3ut−ut+ ∆3u= 0. Appl. Anal. 7, 207–202, 1978.
4. Colton, D. and Wimp, J. Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperature, J. Math. Anal. Appl. 2, 411–418, 1979.
5. Conzalez-Velasco, E. A. Fourier Analysis and Boundary Value Problems (Academic Press, New York, 1995).
6. Halilov, H. On mixed problem for a class of quasilinear pseudoparabolic equation. Appl. Anal. 75 (1-2), 61–71, 2000.
7. Halilov, H. On mixed problem for a class of quasilinear pseudo - parabolic equations. Journal of Kocaeli Univ., Pure and Applied Math. Sec. 3, 1–7, 1996.
8. Hasanov, K. K. On solution of mixed problem for a quasilinear hiperbolic and parabolic equation(PhD. Thesis, Baku, 1961). [9] IL’in, V. A. Solvability of mixed problem for hyperbolic and parabolic equation, Uspekhi Math. Nauk, 15:2, 92, 97–154, 1960. [10] Ladyzhenskaya, D. A. Boundary Value Problems of Mathematical Physics (Springer, New York, 1985).

9. Rao, V. R. and Ting, T. W. F Initial-value problems for pseudoparabolic partial differential equations, Indiana Univ. Math. J. 23, 131–153, 1973.
10. Rundell, W. The solution of initial- boundary value problems for pseudoparabolic partial differential equations, Proc. Roy. Soc. Edin. Sect. A. 74, 311–326, 1975.