Numerical Solution Of High-order ODEs、2D And3D Poisson Equations And Biharmonic Equations Using Haar Wavelets

Wavelet analysis, which is a milestone in the history of Fourier analysis, is a newmathematical branch and developing rapidly. As the wavelet theory into a perfect, ithas been widely applied in many fields. In mathematics, construct fast numerical anal-ysis, numerical methods, curve and surface construction, solve diferential equations,control theory, and so on. Wavelet has many advantages such as regularity, vanishingmoments, compact support and other features, so it is a new research subject to solvediferential equation.Haar wavelets are mathematically the simplest orthogonal wavelets with a compactsupport. Due to the mathematical simplicity Haar wavelet method has turned out tobe an efective tool for solving calculus equations. In the paper Haar wavelet methodfor solution of high-order diferential equations and eigenvalues,2D and3D Poissonequations and biharmonic equations.Haar wavelet method for solving high-order diferential equation is proposed. Haarseries expansion the highest derivative appearing in diferential equation, the otherderivatives(and the function) are obtained through integrations. The constants, whichappeared from integrating process, are calculated from boundary conditions. All thereingredients are then in corporated into the whole system, discrietized by collocationmethod to find solutions. Similarly, we can solve the eigenvalue problems using Matlabcommand EIG.Solve2D Poisson equations and biharmonic equations using tensor product Haarwavelets. Firstly, the horizontal function values are expressed using1D Haar waveletsalong horizontal grid, secondly, the vertical function values are expressed using1DHaar wavelets along vertical grid, finally, the function values of the plane grids are obtained through tensor products. We get numerical solutions by solving the algebraicequations. The method in this paper is feasible and efective through specific examples.As solving3D Poisson equations,6-order partial derivatives are expressed by3DHaar wavelets, and then the integrals of every variable is computed many times. Finalfunction expressions could be obtained. The constants which appeared from inter-grating process are determined by the boundary conditions. The method to solve3Dbiharmonic equations are similar with that to solve3D Poisson equations, and finallyalgebraic systems can be obtained. Although the calculating process is complex, theprecision is high.The method is tested by the help of some numerical examples, for which the exactsolution is known.