| In trying to solve nonlinear partial differential equations with time dependence using the Galerkin method, one ends up with solving nonlinear systems of ordinary differential equations, which are not easily solved. In this thesis we introduce a new iterative method based on eigenfunction expansion to deal with the finite non-linear systems sequentially.; We apply the new method to integrate the semi-linear parabolic equation ut=12u+fu ,x∈D with homogeneous Dirichlet or Robin boundary conditions. We prove the convergence of the new iterative method, and use it to find the multiple solutions of the system, which are difficult to obtain using the Galerkin method.; ; We next apply the new method to solve a parabolic system of two semi-linear equations; ut=12u+f u,q qt=12 q+gu,q ,x∈D with homogeneous boundary conditions Au = 0 and Bv = 0. We prove the convergence of the new method for the case when A = B. If A ≠ B no analytical statements are obtained. However, the proof of convergence is a sufficient, but not a necessary condition, and numerical calculations indicate that the solution obtained by the new method still converges to that obtained by the Galerkin method for the case when A ≠ B. We apply the new method to integrate a system in combustion theory, and we are able to find critical (as defined in Chapter 4) solutions for the system, which are not easily found using the Galerkin method.; To see that the new method can be applied to more general systems, we use it to integrate the Kuramoto-Sivashinsky equation; 6tu+41+e 264xu+e6 2xu+12 6xu2=0 ,x∈0,ℓ We prove the convergence of the iterative method and use it to find the first term of the eigenfunction expansion analytically, and from that we notice that the equation has two solutions, one stable and the other unstable. This kind of observation can not be obtained using the Galerkin method.; Finally, we apply the new method to solve a wave type equation governing the motion of a fluid in a conveying pipe,; EI64w 6x4+&parl0;MU2 t+M6U6t L-x&parr0;6 2w6x2+2MU6 2w6x6t+M +m62w 6t2=0. In all of the above systems, numerical calculations indicate that the solutions obtained by the new method and the Galerkin method coincide. |
