Comparative analysis of solution methods for the telegraph equation: differential transform, reduced differential transform, and second-kind Chebyshev collocation techniques

Abstract This article is about how to solve the telegraph equation using three methods: the differential transform method (DTM), the reduced differential transform method (RDTM), and the second-kind Chebyshev collocation method (SKCCM). Basically, it breaks down how these methods work and what makes them special. The first, DTM, turns differential equations into series solutions and uses some steps to get there. Sometimes, it can even nail down exact answers. The second, RDTM, is like a streamlined version of DTM that makes the math easier. It still deals with series solutions, but without as much heavy lifting. The last, SKCCM, uses Chebyshev polynomials to estimate numerical solutions. It usually turns the differential equation into a bunch of algebraic equations by picking certain points. The paper also looks at how well these methods work and what kind of errors to expect. An analysis of the convergence behavior is also presented in the paper. We solved many test problems using our methods and compared the errors with those obtained from other methods. The results of this comparison highlight the superior accuracy and effectiveness of the proposed techniques over alternative methods.