On numerical solutions of Telegraph, viscous, and modified Burgers equations via Bernoulli collocation method

The presented work aims to develop a novel technique for solving a general form of both linear and nonlinear partial differential equations (PDEs). This technique is based on applying a collocation method with the aid of Bernoulli polynomials and the use of such an algorithm to solve different types of PDEs. The method applies the regular finite difference scheme to convert the model equation into a system of a linear or nonlinear algebraic equation and then this system is solved using a novel iterative technique. Then, by solving this system an unknown coefficient is acquired and an approximate solution for the problems is achieved. Some test results of famous equations including the telegraph, viscous Burger, and modified Burger equations are presented to demonstrate the effectiveness of the proposed algorithm along with a comparison with other related techniques. The method proves to provide accurate results in terms of absolute error and through graphical representation of the solution.