European Journal of Molecular & Clinical Medicine

This paper is devoted to propose the numerical solution pantograph differential equations via a new computational approach of Hermite collocation method. The convergence of the Hermite collocation method is investigated. The numerical solution of pantograph differential equations is obtained in terms of Hermite polynomial. Nonlinear pantograph differential equations are solved and compared with the exact solutions to show the validity, applicability, acceptability and accuracy of the Hermite collocation method. The approximated results show good agreement with the exact solutions, hence indicate good performance of the methods in solving the corresponding equations. 

The non-linear Swift-Hohenberg (SH) equation broadly exists as a model in the study of pattern formation. This equation is in focus of researchers because of the patterns existing in the solution. This equation models many of the fluids dynamics phenomenon and the genetic materials such as tissues in study of the pattern formation. Many of the researchers have obtained the solution of the equation based on different analytical and numerical approaches. This manuscript is dealt with the numerical solution of SH equation using the collocation approach using B-spline basis functions. For the different values of the parameters involved, the equation is studied graphically for the formation of patterns.