This study represents the solution of the Harry Dym equation (HDE) that was calculated using the Lie symmetry group method. This method transforms the controlled partial differential equation with boundary and initial conditions to the problem that containing boundary value of ordinary differential equation. The Chebyshev spectral approximation method is used to solve this problem numerically. This approach is performing by using the highest order derivatives of Chebyshev approximations, as starting steps, followed by getting approximations to the derivatives of lower order. Our obtained numerical results were compared with other works is done. It is seen that the proposed method for the Harry Dym equation gives more accurate results comparing with mentioned works.
The numerical solution of convection-diffusion equation is presented by using Chebyshev spectral method based on El-Gendi method via Lie group analysis. Firstly, we apply Lie symmetry group analysis for the convection-diffusion equation. This method yields convection-diffusion equation to a system of ordinary differential equations (ODEs). Secondly, this system is solved numerically by using Chebyshev spectral method. The numerical results obtained by this way are compared with the exact solution.
In this study, we numerically solve the singular linear ordinary differential equations (SLODEs) of higher order using the collocation method based on the NB1 polynomial. An operational matrix form of the given ordinary differential equations (ODEs), the relations of various solutions and the derivatives are obtained from NB1 polynomials. The proposed method reduces the given problem to a linear algebraic equation system, which removes the singularity of ordinary differential equations. The inverse matrix method is used to solve the resulting system to obtain the coefficients of NB1 polynomials. The obtained exact solutions to different problems of high orders show the reliability and accuracy of the presented method.
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