article · Advances in Continuous and Discrete Models
Solving the telegraph equation requires robust mathematical techniques. Three distinct approaches are analysed: the differential transform method, the reduced differential transform method, and the second-kind Chebyshev collocation method. The differential transform method converts differential equations into series solutions through sequential steps, occasionally producing exact solutions. The reduced differential transform method streamlines this mathematical process, reducing calculation intensity while still delivering series solutions. In contrast, the second-kind Chebyshev collocation method applies Chebyshev polynomials to approximate numerical solutions by converting the governing differential equation into a system of algebraic equations across selected points. Performance evaluations, error analyses, and convergence studies carried out across multiple test problems demonstrate that these three techniques offer superior accuracy and effectiveness compared to existing alternative methods.
The telegraph equation models phenomena such as wave propagation and electrical signal transmission. Developing more accurate and computationally efficient numerical techniques helps researchers reliably simulate complex physical systems. Verifying the convergence and error bounds of these methods provides greater confidence when solving equations where exact physical behaviour must be predicted accurately.
The abstract does not indicate an application pathway or commercial user base for these mathematical methods.
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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.
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DOI: 10.1186/s13662-026-04123-x
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