RUDN University mathematician suggested a scheme for solving telegraph equations

February 11, 2021

A mathematician from RUDN University suggested a stable difference scheme for solving inverse problems for elliptic-telegraph and differential equations that are used to describe biological, physical, and sociological processes. The results of the study were published in the Numerical Methods for Partial Differential Equations journal.

Elliptic equations are a class of differential equations in partial derivatives that are used, among other things, to model time-independent processes. Telegraph equations are presented in a nonstationary form. They were initially obtained for a telegraph communication line, but today they are also used to model the movement of insects, the flow of blood through veins, and the changes underwent by building materials. Moreover, they can be inversed, i.e. used to find a source of changes based on known process characteristics, for example, to identify a cause of material damage or to create an optic tomography image for the purposes of medical diagnostics. It is often difficult to obtain accurate solutions for problems like that, therefore, the initial problem is reduced to a system of simpler equations that provide an answer with a certain degree of approximation to the correct one. A mathematician from RUDN University suggested an algorithm to obtain inverse problem solutions for elliptic-telegraph equations using a computer.

"The more complex a modeled system, the more unknown parameters it contains, and the more difficult are the calculations. However, despite the complexity of the task, modern computers can be used to search for approximate solutions to differential equations. We aimed to obtain absolute stable difference schemes for the approximate solution of the space identification problem for the elliptic?telegraph equations. Our work could help further implement these methods into the modeling of various processes," said Prof. Allaberen Ashyralyev, a PhD in Physics and Mathematics from the Department of Higher Mathematics, RUDN University.

One way to obtain an approximate solution is to replace the initial problem with difference schemes. The studied area is turned into a grid with a given step size, and functions are replaced with nod values. The mathematician suggested a difference scheme and then studied it both analytically and numerically. The first method was used to confirm the absolute stability of the scheme, and the second (a numerical experiment, i.e. an equation that the scheme was applied to) - to support the results of the analysis. The scientist managed to demonstrate that the scheme was absolute stable and independent from the chosen calculation step size.

"Similar elliptic-telegraph equations are used to model biological systems, sociological phenomena, and engineering processes. An absolute stable difference scheme could help specialists better study these issues," added Prof. Allaberen Ashyralyev from RUDN University.

RUDN University

Related Mathematician Articles from Brightsurf:

RUDN University mathematician refined the model of predator-prey relations in the wild
The traditional mathematical model of predator-prey relations in the wild does not take into account indirect nonlocal interactions.

NUI Galway mathematician publishes article in world's top mathematics journal
An Irish mathematician, Dr Martin Kerin, from the School of Mathematics, Statistics and Applied Mathematics at NUI Galway, has had a research article published in the Annals of Mathematics, widely regarded as the top journal for pure mathematics in the world.

A method for predicting antiviral drug or vaccine targets
A novel method to predict the most promising targets for antiviral drugs or vaccines is based on the conformational changes viral glycoproteins go through during the process of recognition and binding to the host cell.

French mathematician and spider aficionado Cédric Villani honoured with a new orb-weaver
Considered as one of the best studied spiders, the orb-weavers remain poorly known in the central parts of the Palearctic ecozone.

UT mathematician develops model to control spread of aquatic invasive species
Adjusting the water flow rate in a river can prevent invasive species from moving upstream and expanding their range.

RUDN University mathematician first described the movement in a flat strip of plasma
RUDN University mathematician for the first time proved the theorem of existence and uniqueness of solutions of the Zakharov-Kuznetsov equation in a strip.

The ever-winning lottery ticket: Mathematicians solve a dusty mystery
After years of work, University of Copenhagen mathematics researchers have answered a mysterious half-century-old riddle.

People can see beauty in complex mathematics, study shows
Ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a beautiful landscape painting or a piano sonata.

WPI mathematician is helping NASA spacecraft travel faster and farther
By combining cutting-edge machine learning with 19th-century mathematics, a Worcester Polytechnic Institute (WPI) mathematician is working to make NASA spacecraft lighter and more damage tolerant by developing methods to detect imperfections in carbon nanomaterials used to make composite rocket fuel tanks and other spacecraft structures.

Sussex mathematician's breakthrough on non-toxic pest control
Breakthrough 'gene silencing' technique uses naturally occurring soil bacteria to kill specific crop-destroying pests without harming other insects or the environment.

Read More: Mathematician News and Mathematician Current Events is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to