JADEs Issue #8: Wave & Telegraph PDEs

In this week's issue, we explore or discuss the following homogeneous hyperbolic linear PDE, maybe the most famous one of them all... the Wave Equation. And, later, we discuss a Telegraph PDE, also an equation of hyperbolic type. But, first, we dive into the Wave Equation.

PROBLEM

Definition

Given the following homogeneous Wave Equation:

With the following conditions on the boundary:

We get the following traveling wave solution u = u(x,t):

Source: Polyanin et al, Handbook of Linear PDEs for Engineers and Scientists, CRC Press (2012).

Example

With a = 3 in this form of the Wave Equation, we get:

On these boundary conditions:

Then, the traveling wave solution u = u(x,t) becomes the following:

Our next problem deals with the Telegraph PDE. This time, it's a nonlinear PDE of hyperbolic-type.

PROBLEM 2 (EXCERPT)

Definition

Given the following nonlinear Telegraph PDE:

You'll get the following general solution:

Example

Source: Anglin, Telegraph PDEs, Kindle Direct (2022).

ABOUT THE JOURNAL

The Journal of Applied Differential Equations, otherwise known as JADEs is a free Open Access (OA) eJournal (ISSN# pending) that covers both linear and nonlinear partial differential equations (PDEs) and ordinary differential equations (ODEs) and the analytic methods/solutions to these.

While most journals and proceedings take a very theory-driven approach to PDEs and ODEs, this journal will discuss and illustrate problems from a very pragmatic, workshop problem-solution focus for applied/industrial mathematicians, scientists (physicists, chemists, etc.), engineers (mechanical, electrical, maybe some structural), and others.

Please subscribe if you want to see more problems like the ones above as well as other more relevant PDEs and/or ODEs with applications in mechanics (analytical, classical, Newtonian), electromagnetic field/quantum-based wave mechanics, fluid mechanics and beyond. Enjoy!

ABOUT THE EDITOR

Steve Anglin, M.Sc. Ph.D. (h.c.) is an applied, industrial mathematician primarily teaching and solving partial differential equations (PDEs) with applications in the fields of fluid mechanics (mechanical engineering) as well as some electrical engineering and physics. He is a former lecturer of mathematics at Case and Saint Leo Universities. Steve received his Master of Science (M.Sc.) in applied mathematics from Brown University (Ivy League) and his Hon. Doctorate (Ph.D.(h.c.)) in mathematics from Trinity College. Previously, he published or has been credited on 2,000+ technical trade books with Springer Nature and Pearson Education as well as 2 eZines with O'Reilly Media.

For more, visit:?https://www.amazon.com/author/steveanglin

Email:?[email protected]