A haar wavelet series solution of heat equation with involution

It is well known that the wavelets have widely applied to solve mathematical problems connected with the differential and integral equations. The application of the wavelets possess several important properties, such as orthogonality, compact support, exact representation of polynomials at certain degree and the ability to represent functions on different levels of resolution. In this paper, new methods based on wavelet expansion are considered to solve problems arising in approximation of the solution of heat equation with involution. We have developed new numerical techniques to solve heat equation with involution and obtained new approximative representation for solution of heat equations

Numerical solution of the inverse Gardner equation

In this paper, the numerical solution of the inverse Gardner equation will be considered. The Haar wavelet collocation method (HWCM) will be used to determine the unknown boundary condition which is estimated from an over-specified condition at a boundary. In this regard, we apply the HWCM for discretizing the space derivatives and then use a quasilinearization technique to linearize the nonlinear term in the equations. It is proved that the proposed method has the order of convergence O(∆x). The efficiency and robustness of the proposed approach for solving the inverse Gardner equation are demonstrated by one numerical example.Publisher's Versio

Application of higher order Haar wavelet method for solving nonlinear evolution equations

The recently introduced higher order Haar wavelet method is treated for solving evolution equations. The wave equation, the Burgers’ equations and the Korteweg-de Vries equation are considered as model problems. The detailed analysis of the accuracy of the Haar wavelet method and the higher order Haar wavelet method is performed. The obtained results are validated against the exact solutions