Analytical and numerical study on a variable scalar equation

Basically, scalar equations have potential applications in various fields such as the transmission of nerve impulses between neurons through myelin substance and other disciplines. A particular model is well-known as the pantograph equation. The standard version of this scalar equation has been extensively investigated via different analytical and numerical techniques. This paper considers a variable version of the pantograph equation. Usually, constructing an exact or a closed form solution for a variable scalar equation is a challenge. However, this work proposes a developed hybrid approach to overcome such a difficulty. The solution of the current variable version is analytically obtained in different closed forms with addressing the convergence criteria. Under some conditions, such closed forms are successfully converted to different exact ones. Additionally, accurate approximations are provided and examined. Several comparisons with the available exact solutions are conducted as a validation of our approximations. Besides, the accuracy of our approximations is checked for some classes which have no exact solutions. Probably, the results demonstrate the elegance of the proposed approach to deal with a variable version of the Pantograph model.