Approximate Solution of Mixed Problem for Telegrapher Equation with Homogeneous Boundary Conditions of First Kind Using Special Functions

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

1. Heaviside O. (1951) Electromagnetic Theory. 3rd ed. London, Spon. 416.

2. Ango A. (1964) Mathematics for Electrical and Radio Engineers. Мoscow, Nauka Publ. 772 (in Russian).

3. Koshlyakov N. S., Gliner E. B., Smirnov M. M. (1962) Differential Equations of Mathematical Physics. Мoscow, GIFML. 767 (in Russian).

4. Aramanovich I. G., Levin V. I. (1969) Equations of Mathematical Physics. Мoscow, Nauka Publ. 288 (in Russian).

5. Smirnov V. I. (1974) Course on Higher Mathematics. Vol. 2. Мoscow, Nauka Publ. 479 (in Russian).

6. Myshkis A. D. (2007) Lectures on Higher Mathematics. Saint-Petersburg, Lan’ Publ. 688 (in Russian).

7. Ostapenko V. (2012) Telegrapher's Equation. Boundary Value Problems. Saarbrücken, LAP Lambert Academic Publ. 272 (in Russian).

8. Novikov Yu. N. (2005) Electrical Engineering and Electronics. Theory of Circuits and Signals, Methods of Analysis. Saint-Petersburg, Piter Publ. 384 (in Russian).

9. Bychkov Yu. A., Zolotnitskii V. M., Chernyshev E. P. (2002) Fundamentals of the Theory of Electrical Circuits. Saint-Petersburg, Lan’ Publ. 464 (in Russian).

10. Dubnishchev Yu. N. (2011) Vibrations and Waves. Saint-Petersburg, Lan’ Publ. 384 (in Russian).

11. Lasy P. G., Meleshko I. N. (2017) Approximate Solution of One Problem on Electrical Oscillations in Wires with the Use of Polylogarithms. Energetika. Izvestiya Vysshikh Uchebnykh Zavedenii i Energeticheskikh Ob’edinenii SNG = Energetika. Proceedings of CIS Higher Education Institutions and Power Engineering Associations, 60 (4), 334–340. https://doi.org/10.21122/1029-7448-2017-60-4-334-340 (in Russian).

12. Lasy P. G., Meleshko I. N. (2019) Application of Polylogarithms to the Approximate Solution of the Inhomogeneous Telegraph Equation for the Distortionless Line. Energetika. Izvestiya Vysshikh Uchebnykh Zavedenii i Energeticheskikh Ob’edinenii SNG = Energetika. Proceedings of CIS Higher Education Institutions and Power Engineering Associations, 62 (5), 413–421. https://doi.org/10.21122/1029-7448-2019-62-5-413-421 (in Russian).

13. Pykhteev G. N., Meleshko I. N. (1976) Polylogarithms, their Properties and Calculation Methods. Minks, BSU Publ. 68 (in Russian).

14. Meleshko I. N. (1999) Special Formulas for Cauchy-Type Integrals and their Applications. Minks, VUZ-YUNITI Publ. 197 (in Russian).

15. Fikhtengol'ts G. M. (1969) Course on Differential and Integral Calculus. Vol. 3. Мoscow, Nauka Publ. 656 (in Russian).

16. Fikhtengol'ts G. M. (1962) Course on Differential and Integral Calculus. Vol. 2. Мoscow, Fizmatgiz Publ. 807 (in Russian).