Hyperbolic partial differential equation

• Elkholy
In summary, the general solution to the hyperbolic partial differential equation given is a combination of wave-like solutions and separable solutions, where the functions must satisfy certain conditions based on the constants a, b, c, and d. These solutions can be tested by plugging in initial conditions to determine if they satisfy the equation.
Elkholy
What is the general solution of the following hyperbolic partial differential equation:

The head (h) at a specified distance (x) is a sort of a damping function in the form:

Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance).

Well, just looking at it, you know it has to be exponential in t. If we guess ##h_1=e^{-a*t}##, that works as one homogeneus solution, although the second 2 terms become arbitrary. So h_1 is part of the general solution, we just need other solutions to make the second 2 terms non arbitrary. So the second derivative with respect to x must differ from partial x partial t only by a constant. x works, as it's derivative is zero for all partials above, so far we have ##h_{1+2}=k_1e^{-a*t}+k_2x## where ##k_{n}## is an arbitrary constant. Now you try to throw a few out there, I have a feeling there are a lot, or at least one more function of x and t. I honestly don't know though.

BiGyElLoWhAt said:
Well, just looking at it, you know it has to be exponential in t. If we guess ##h_1=e^{-a*t}##, that works as one homogeneus solution, although the second 2 terms become arbitrary. So h_1 is part of the general solution, we just need other solutions to make the second 2 terms non arbitrary. So the second derivative with respect to x must differ from partial x partial t only by a constant. x works, as it's derivative is zero for all partials above, so far we have ##h_{1+2}=k_1e^{-a*t}+k_2x## where ##k_{n}## is an arbitrary constant. Now you try to throw a few out there, I have a feeling there are a lot, or at least one more function of x and t. I honestly don't know though.

I know that the governing differential equation of a free vibration of a damped system has this form

and its solution (subject to initial conditions) is

where,
, ω_n is the natural frequency and ζ is a damping factor <1 and if drawn it gives

But, the problem here is the third and the fourth terms.
Is it right to just add this term k_2 x to the previous solution?

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The all cancel out. The general solution is the sum of all homogenous solutions (they all equal zero). This isn't an enveloped oscillator, at least as far as I can see.

Basically, the general solution to this differential equation is any terms that you can add to gether that, when you plug the whole thing into this PDE, equals zero. I will almost guarantee that there are more terms. One way you can check is by taking 2 initial conditions and plugging them into the function, see if this function satisfies both conditions, if not, then we're missing stuff.

Elkholy said:
What is the general solution of the following hyperbolic partial differential equation:
View attachment 85690
The head (h) at a specified distance (x) is a sort of a damping function in the form:
View attachment 85691
Where, a, b, c and d are constants. And the derivatives are with respect to t (time) and x (distance).

There are wave-like solutions $h(x,t) = f(x - vt)$ where $f$ must satisfy $$(v^2 - bv + c)f'' - av f' = 0.$$

There are also separable solutions of the form $h(x,t) = X(x)e^{-kt}e^{i\omega t}$ where $X$ must satisfy $$cX'' + b(-k + i\omega)X' + ((-k + i\omega)^2 + a(-k + i\omega))X = 0.$$

hunt_mat

1. What is a hyperbolic partial differential equation?

A hyperbolic partial differential equation is a type of mathematical equation that describes the behavior of a system over time, taking into account both spatial and temporal variables. It is called "hyperbolic" because its solutions often involve hyperbolic functions.

2. How is a hyperbolic partial differential equation different from other types of PDEs?

A hyperbolic PDE is characterized by having two distinct families of characteristic curves, which are curves along which the solution is constant. This is different from elliptic PDEs, which have no characteristic curves, and parabolic PDEs, which have only one family of characteristic curves.

3. What are some real-world applications of hyperbolic PDEs?

Hyperbolic PDEs have a wide range of applications in various fields such as fluid dynamics, electromagnetism, and general relativity. They are used to model phenomena such as sound waves, electromagnetic waves, and gravitational waves.

4. How do you solve a hyperbolic PDE?

Solving a hyperbolic PDE involves finding a solution that satisfies both the given equation and any initial or boundary conditions. This can be done using analytical methods, such as the method of characteristics, or numerical methods, such as finite difference or finite element methods.

5. What are the limitations of hyperbolic PDEs?

One limitation of hyperbolic PDEs is that they can only accurately model systems that are linear or weakly nonlinear. They also have difficulty accounting for certain physical phenomena, such as shock waves, which require more complicated models. Additionally, solving hyperbolic PDEs can be computationally expensive and time-consuming.

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