# I can't find anywhere how to solute PDE's.

• lukaszh
In summary, the conversation discusses techniques for solving partial differential equations, specifically using separation of variables. The process involves factoring the equation into two one-variable functions and then finding constants that satisfy certain conditions. This technique can be used to find a general solution, which can then be modified to fit specific boundary conditions.
lukaszh
I can't find anywhere how to solute PDE's. For exaple ODE:
$$\frac{du}{dt}=u\Rightarrow\ln u=t+\ln C\Rightarrow u=Ce^t$$
But this?
$$\frac{du}{dt}+\frac{du}{dx}=u$$

$$dudx+dudt=udtdx\Rightarrow udx+udt=udtdx$$

I haven't had pde's yet, but I'm interested in solving these equations :-(

For many equations, there is a technique that works called "separation of variables". To separate the equation, we assume u(x,t) can be factored into two one-variable functions:

$$u(x,t) = X(x)T(t)$$

And then

$$\frac{\partial u}{\partial x} = T(t)\frac{dX(x)}{dx}$$

$$\frac{\partial u}{\partial t} = X(x)\frac{dT(t)}{dt}$$

Then, for your equation, we can write:

$$X(x)\frac{dT(t)}{dt} + T(t)\frac{dX(x)}{dx} = X(x)T(t)$$

Next, divide by X(x)T(t):

$$\frac{1}{T(t)}\frac{dT(t)}{dt} + \frac{1}{X(x)}\frac{dX(x)}{dx} = 1$$

Now, notice that we've separated the variables into the form

$$P(t) + Q(x) = 1$$

Since this must be true for all x and t, the functions P and Q must individually be constant. This gives

$$a + b = 1$$

$$\frac{1}{T(t)}\frac{dT(t)}{dt} = a$$

$$\frac{1}{X(x)}\frac{dX(x)}{dx} = b$$

These have the solution

$$X(x) = e^{bx}$$

$$T(t) = e^{at}$$

and so

$$u(x,t) = Ce^{at}e^{bx} = Ce^{at + (1-a)x}$$

for arbitrary constants C and a. Also note that any linear combination of u's (with different values for a) are also solutions, because the equation is linear. This can be used to construct a more general solution, given some boundary conditions.

## 1. What are PDEs and why are they important in science?

PDEs, or partial differential equations, are mathematical equations that involve multiple variables and their partial derivatives. They are important in science because they are used to model real-world phenomena in fields such as physics, engineering, and economics.

## 2. How do I know if a problem can be solved using PDEs?

Problems that involve multiple variables and their rates of change, such as heat transfer, wave propagation, and diffusion, can often be solved using PDEs. It is important to consult with a mathematician or expert in the specific field to determine if PDEs are the appropriate tool for a particular problem.

## 3. Are there different types of PDEs?

Yes, there are several types of PDEs, including elliptic, parabolic, and hyperbolic equations. These types differ in the way they model physical phenomena and the methods used to solve them.

## 4. What is the process for solving a PDE?

The process for solving a PDE can vary depending on the type and complexity of the equation. Generally, it involves using techniques such as separation of variables, Fourier transforms, and numerical methods to find a solution. It is important to have a strong understanding of calculus and differential equations to effectively solve PDEs.

## 5. Can PDEs be solved analytically or only numerically?

Some PDEs can be solved analytically, meaning a closed-form solution can be obtained. However, many PDEs are too complex to be solved analytically and require numerical methods, such as finite difference or finite element methods, to obtain an approximate solution.

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