scholarly journals A WAVELET-TAYLOR GALERKIN METHOD FOR PARABOLIC AND HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS

2005 ◽  
Vol 02 (01) ◽  
pp. 75-97 ◽  
Author(s):  
B. V. RATHISH KUMAR ◽  
MANI MEHRA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Linear Differential Operator ◽  
Nonlinear Function ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Taylor Series Expansions ◽  
Time Marching ◽  
Linear Differential

In this study a set of new space and time accurate numerical methods based on different time marching schemes such as Euler, leap-frog and Crank-Nicolson for partial differential equations of the form [Formula: see text], where ℒ is linear differential operator and [Formula: see text] is a nonlinear function, are proposed. To produce accurate temporal differencing, the method employs forward/backward time Taylor series expansions including time derivatives of second and third order which are evaluated from the governing partial differential equation. This yields a generalized time discretized scheme which is approximated in space by Galerkin method. The compactly supported orthogonal wavelet bases developed by Daubechies are used in Galerkin scheme. This new wavelet-Taylor Galerkin approach is successively applied to heat equation, convection equation and inviscid Burgers' equation.

scholarly journals New Approach for Solving Partial Differential Equations Based on Collocation Method

2020 ◽  
Vol 18 ◽  
pp. 118-128
Author(s):  
Alaa Almosawi ◽  
Luma N. M. Tawfiq
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Collocation Method ◽  
Linear Differential Operator ◽  
Partial Differential ◽  
Restrictive Assumption ◽  
New Approach ◽  
Linear Differential

In this paper, a new approach for solving partial differential equations was introduced. The collocation method based on LA-transform and proposed the solution as a power series that conforming Taylor series. The method attacks the problem in a direct way and in a straightforward fashion without using linearization, or any other restrictive assumption that may change the behavior of the equation under discussion. Five illustrated examples are introduced to clarifying the accuracy, ease implementation and efficiency of suggested method. The LA-transform was used to eliminate the linear differential operator in the differential equation.

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