Spline Solutions For Four Order Parabolic Equations Based On The Sub-domain Precise Integration Method

In this dissertation, the new methods based on sub-domain precise integration and non-polynomial spline for solving the four order parabolic equation are presented. The maincontents of the dissertation are as follows:1. First of all, the existing methods and results for four order parabolic equation arepresented.2. The n degree univariate spline functions, the basic idea for solving differential equa-tions based on quintic spline, fifth-degree B-spline function and some related basic relation-ships of quintic spline and parametric quintic spline are brieffy introduced.3. The development of precise integration method and the application of sub-domainprecise integration method for solving four order parabolic equation is presented.4. The application of the fifth-degree B-spline function for solving four order parabolicequation is introduced firstly; next, the new numerical solution based on sub-domain pre-cise integration is presented; finally, the numerical examples show that the present method isapplicable and approximates the exact solution very well.5. By using the sub-domain precise integration method in time and the quintic spline inspace, a five-point and two level implicit scheme for solving the periodic initial value problemof four order parabolic equation is presented. It is shown that the scheme is unconditionallystable and the local truncation error is O (α△t + (△t)~2 + (△x)~2). Some numerical examples aretested.6. By using the sub-domain precise integration method and non-polynomial quinticspline, a five-point and two level implicit scheme for solving the periodic initial value prob-lem of four order parabolic equation is presented. Next, the stability and truncation error areanalyzed and found that this scheme is unconditionally stable and the local truncation erroris O (α△t + (△t)~2 + (△x)~6). It has shown by numerical results that the new scheme is effectiveand practicable.