Utilization of Chebyshev collocation approach for differential, differential-difference and integro-differential equations

Abstract

Numerical solutions to differential and integral equations have been a very active field of research. The necessary tools to solving differential equations can add much fuel for acceleration of scientific development. Therefore, it is quite challenging for scientists, especially mathematicians, to work on the solution methodologies for differential equations. In this study, we have discussed the use of Chebyshev collocation method for solving a large variety of problems including various types of integral equations, integro-differential equations, and differential-difference equations. The method under consideration relies on a trial solution which is a linear combination of basic functions derived from Chebyshev polynomials. The trial solution is then used to obtain a residue function whose vanishing weighted residual can be used to obtain a system of algebraic equations which gives the values of constants used in the trial solution. Plugging the constants back into the trial solution, we obtain an approximate solution to the problem at hand. We have provided the comparison between exact and approximate solutions for different problems and found an exceptionally good agreement in results. The comparison has been provided through plots and tables to portray the accurateness of Chebyshev collocation method. The behaviour of the solutions is also displayed with the help of the graphs.

Keywords

• collocation approach
• mathematical modelling
• numerical solution
• differential equation
• integro-differential equation
• differential-difference equation