# Verifying if this PDE is a solution

• jc2009
In summary, the conversation discusses verifying the functions [x+1]e^(-t), e^(-2)sint, and xt as solutions to nonhomogeneous equations involving the 1D heat operator H = \frac{\partial}{\partial t} - \frac{\partial^2}{\partial x^2}, and finding a solution for the PDE Hu = \sqrt{2} x + [Pi]e^(-2x) [4sint + cost] + e^(-t)[x+1]. It is noted that the DE is linear nonhomogeneous and that if u1(x,t) and u2(x,t) are solutions to Hu =\phi_1(x,t) and Hu
jc2009
PROBLEM: Verify that the functions [x+1]e^(-t) ; e^(-2)sint ; and xt are respectively solutions of the nonhomogeneous equations
Hu = -e^(-t)[x+1] ; Hu = e^(-2x)[4sint+cost] ; and Hu = x
where H is the 1D heat operator H = $$\frac{\partial}{\partial t}$$ - $$\frac{\partial^2}{\partial x^2}$$

i did this the verification part,, the problem is with the second part of the problem
Find a solution of the PDE
Hu = $$\sqrt{2} x$$ + [Pi]e^(-2x) [4sint + cost] + e^(-t)[x+1]

isn't the first part a solution for this PDE? i don't understand the question

any hints how to setup this PDE?

You have already verified some particular solutions for $$Hu =\phi(x,t).$$
Note that the DE is linear nonhomogeneous.

If u1(x,t) (resp. u2(x,t)) is a solution of $$Hu =\phi_1(x,t)$$ (resp. $$Hu =\phi_2(x,t)$$ )
then
u1(x,t) + u2(x,t) will be a solution of

$$Hu =\phi_1(x,t) + \phi_2(x,t).$$

## 1. What is a PDE?

A PDE, or partial differential equation, is an equation that involves functions of multiple variables and their partial derivatives. It is commonly used to model physical phenomena in fields such as physics, engineering, and economics.

## 2. How do you verify if a PDE is a solution?

To verify if a PDE is a solution, you must first substitute the proposed solution into the equation and then check if it satisfies the equation for all values of the variables. This can be done algebraically or numerically.

## 3. What are some common methods for verifying PDE solutions?

Some common methods for verifying PDE solutions include separation of variables, method of characteristics, and the Laplace transform method. These methods involve manipulating the PDE to make it easier to solve or applying specific techniques to find a solution.

## 4. Can a PDE have multiple solutions?

Yes, a PDE can have multiple solutions. In fact, most PDEs have an infinite number of solutions. This is because there are often multiple ways to satisfy the equation for a given set of initial or boundary conditions.

## 5. Are there any software tools that can help verify PDE solutions?

Yes, there are many software tools that can help verify PDE solutions. Some popular options include Mathematica, MATLAB, and Python libraries such as SymPy and SciPy. These tools use powerful algorithms to solve PDEs and verify their solutions.

Replies
5
Views
579
Replies
1
Views
1K
Replies
3
Views
1K
Replies
4
Views
2K
Replies
3
Views
2K
Replies
5
Views
2K
Replies
4
Views
784
Replies
13
Views
2K
Replies
3
Views
1K
Replies
2
Views
857