Partial differential equation with conditions

In summary: The boundary conditions will be satisfied automatically. In summary, the conversation discusses a differential equation and its conditions, and the suggestion to use the steady state solution to get a homogeneous initial condition. The idea of separating variables is introduced, and the method of substitution is used to make the boundary conditions homogeneous. The final steps involve solving the system with separation of variables to obtain the solution for u(x,t) that satisfies all the conditions.
  • #1
selzer9
2
0
I'm not sure how to solve this:

du/dt = 3 [itex]\frac{d^{2}u}{dx^{2}}[/itex]

These are the conditions:
u(0,t)= -1
u(pi,t)= 1
u(x,0) = -cos 7x

Suggestion:
I should use steady state solution to get a homogeneous initial condition.

Starting with separtion of variables

u(x,t) = G(x)H(t)

And du/dt = GH' and d^2u/dx^2 = G"H

So GH'= c^2 G"H

and H'/(c^2)H = G"/G

Then H'/(c^2)H = k = G"/G

thus H' - (c^2)Hk = 0 and G" - Gk = 0.

Once here how do I make sure u(x,t) follows the conditions?
 
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  • #2
selzer9 said:
I'm not sure how to solve this:

du/dt = 3 [itex]\frac{d^{2}u}{dx^{2}}[/itex]

These are the conditions:
u(0,t)= -1
u(pi,t)= 1
u(x,0) = -cos 7x

Suggestion:
I should use steady state solution to get a homogeneous initial condition.

Starting with separtion of variables

u(x,t) = G(x)H(t)

And du/dt = GH' and d^2u/dx^2 = G"H

So GH'= c^2 G"H

and H'/(c^2)H = G"/G

Then H'/(c^2)H = k = G"/G

thus H' - (c^2)Hk = 0 and G" - Gk = 0.

Once here how do I make sure u(x,t) follows the conditions?

I wouldn't start it that way. You need to make the boundary conditions homogeneous before you separate the variables. I would start with the substitution$$
u(x,t) = v(x,t) + \Psi(x)$$ Substituting that into your system gives$$
v_t = 3v_{xx}+3\Psi''(x)$$ $$
v(0,t) + \Psi(0)= -1$$ $$
v(\pi, t) +\Psi(\pi) = 1$$ $$
v(x,0) + \Psi(x) = -\cos x$$
Now if you let ##3\Psi''(x) = 0,\, \Psi(0) = -1,\, \Psi(\pi)= 1##, you can solve for ##\Psi(x)## and you are left with the homogeneous system$$
v_t = 3v_{xx}$$ $$
v(0,t) = 0$$ $$
v(\pi, t) = 0$$ $$
v(x,0) = -\cos x-\Psi(x)$$Solve this system in the usual way with separation of variables. Once you know ##v(x,t)##, your solution will be ##u(x,t)=v(x,t)+\Psi(x)##.
 
Last edited:

1. What is a partial differential equation?

A partial differential equation (PDE) is a mathematical equation that involves partial derivatives of a function of two or more independent variables. It is used to describe the relationship between the function and its variables, and is commonly used in physics and engineering to model complex systems.

2. What are the conditions in a partial differential equation?

The conditions in a PDE refer to the constraints or limitations that are applied to the solution of the equation. These can include initial conditions, boundary conditions, or any other requirements that must be satisfied by the solution.

3. How is a partial differential equation solved?

The process of solving a PDE involves finding a function that satisfies the equation and its conditions. This can be done analytically, using mathematical techniques such as separation of variables, or numerically, using computational methods like finite difference or finite element methods.

4. What are some real-life applications of partial differential equations?

PDEs are used to model a wide range of physical phenomena, including heat transfer, fluid flow, and wave propagation. They are also used in economics, biology, and other fields to understand complex systems and make predictions.

5. Are partial differential equations difficult to understand?

The complexity of PDEs can vary depending on the specific equation and its conditions, but they can be challenging to understand without a strong mathematical background. However, with practice and a solid understanding of the underlying principles, they can be tackled effectively.

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