Solving a partial differential equation

In summary, the conversation discusses solving a wave equation with a forced-oscillation term on the right-hand side. The speakers mention attempting to solve it in two dimensions and using Mathematica, but no results were obtained. Eventually, changing variables to ##\xi = z-t## and ##\eta = z+t## led to a solvable equation.
  • #1
Haorong Wu
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Homework Statement
##(\partial_t^2-\partial_z^2) h(t,z)=A \cos (k(t-z))##
Relevant Equations
None
If the right-hand side is zero, then it will be a wave equation, which can be easily solved. The right-hand side term looks like a forced-oscillation term. However, I only know how to solve a forced oscillation system in one dimension. I do not know how to tackle it in two dimensions.

I have tried to generalize it into two dimensions by solving it pretending ##h## depends only on ##t## and ##z## separately, but I have no clues on how to carry on.

I have tried it in Mathematica. It gives no results.

Thanks ahead.
 
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  • #2
Try changing variables to ##\xi = z-t## and ##\eta = z+t##.
 
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  • #3
This is special case of the inhomogeneous wave equation or wave equation with source term. The so called source term is the right hand side. If the right hand side is zero, we have the homogeneous wave equation or simply wave equation.
 
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  • #4
Thanks, @Orodruin and @Delta2.

By changing variables with ##t-z=\alpha## and ##t+z=\beta##, I found that the equation becomes ## \partial_\alpha \partial_\beta=\frac A 4 \cos (k\alpha)##, which can be easily solved.
 
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FAQ: Solving a partial differential equation

1. What is a partial differential equation (PDE)?

A PDE is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe physical phenomena in which the rate of change of a quantity is dependent on multiple factors.

2. How do you solve a PDE?

Solving a PDE involves finding a function or set of functions that satisfy the equation. This can be done through various methods such as separation of variables, using integral transforms, or numerical methods.

3. What are the applications of PDEs?

PDEs have a wide range of applications in fields such as physics, engineering, economics, and biology. They are used to model and analyze various phenomena such as heat transfer, fluid dynamics, population growth, and financial markets.

4. What are the differences between ordinary differential equations (ODEs) and PDEs?

The main difference between ODEs and PDEs is the number of independent variables. ODEs involve only one independent variable, while PDEs involve multiple independent variables. Additionally, the solution to an ODE is a function, whereas the solution to a PDE is a function of multiple variables.

5. Are there any real-life examples of PDEs?

Yes, there are many real-life examples of PDEs. Some common examples include the heat equation, which describes the flow of heat in a solid object, and the wave equation, which describes the propagation of waves in a medium. PDEs are also used in image and signal processing, weather forecasting, and in the design of aircraft wings.

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