Legendre wavelet method for numerical solutions of partial differential equations

Abstract

We introduce an orthogonal basis on the square [−1, 1] × [−1, 1] generated by Legendre polynomials on [−1, 1], and define an associated expression for the expansion of a Riemann integrable function. We describe some properties and derive a uniform convergence theorem. We then present several numerical experiments that indicate that our methods are more efficient and have better convergence results than some other methods.

New Implementation of Legendre Polynomials for Solving Partial Differential Equations

“…Table 3. Maximum error and CPU time of present method and Liu-Lin method [7] are listed in Table 4. By comparing the data in Tables 3 and 4, it is clear that our method is more efficient.…”

“…Analytical solution of PDEs , however , either does not exist or is difficult to find . Recent contribution in this regard includes meshless methods [3], finite-difference methods [4], Alternating-Direction Sinc-Galerkin method (ADSG) [5] , quadratic spline collocation method (QSCM) [6] , Liu and Lin method [7] and so on .…”

Legendre Wavelet Expansion of a Function f(x, y) and its Approximation

“…Legendre wavelets possess the orthogonality property. Some results concerning to Legendre wavelet and Haar wavelet have been discussed by researchers Islam [3], Lal and Kumar [5], Lal and Kumar [6], Nanshan [4] and Razzaghi [2] etc. The Legendre wavelet approximation of functions of two variables have not been discussed so far.…”

Fractional Polynomial Method for Solving Fractional Order Population Growth Model

“…For example, the fractional derivatives could have modeled the nonlinear oscillation of earthquake [6]. The fluid-dynamic models with fractional derivatives [10], [11] can eliminate the deficiency arising from the assumption of continuum traffic flow and differential equations with fractional order have recently proved to be valuable tools for the modeling of many physical phenomena [2], [9]. Mainardi [9] discussed the applications of fractional calculus in statistical mechanics and continuum.…”

“…The fluid-dynamic models with fractional derivatives [10], [11] can eliminate the deficiency arising from the assumption of continuum traffic flow and differential equations with fractional order have recently proved to be valuable tools for the modeling of many physical phenomena [2], [9]. Mainardi [9] discussed the applications of fractional calculus in statistical mechanics and continuum. Nowadays fractional differential equations have proved to be valuable tools in the modeling of physical phenomena [1], [2], [4], [5], [13], [15], [17].…”