Partial differential equations with laplas

In summary, To solve the partial differential equation using Laplace transforms, first take the Laplace of the equation and then convert it into a differential equation form using a method that is known to be solvable. One possible method is to get the equation into the form of r^2 - s/(a^2)=0. After taking the Laplace of the equation, using separation of variables can help to solve the equation easily. This involves assuming that u is in the form of u(x,t) = X(x)T(t) and then substituting it back into the original equation to separate it into two second order linear differential equations.
  • #1
jd1828
44
0

Homework Statement



Solve the partial differential equation using laplas transforms:

U`(x)=a^2*U``(t)

given U(x,0)=2


There are more initial conditions but i am just trying to get to the general solution

The Attempt at a Solution



First take laplas of the equation. Then I am trying to get it into a differential equation form that i know how to solve.

im not sure what method its called but I get

r^2 - s/(a^2)=0

Im not to sure if that is correct.
 
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  • #2
I should add that after taking the laplas of the equation I get:

S*U(s)-2=a^2*(d^2/dx^2)U(s)
 
  • #3
solve this using separation of variables in which you assume that u is

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

once you substitute this back int othe original equation you can separate that into two second order lienar DEs which you can solve easily.
 

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

A partial differential equation is a mathematical equation that involves multiple independent variables and their partial derivatives. It describes the relationship between these variables and their rates of change. In the context of Laplace's equation, the PDE involves the Laplace operator, which represents the sum of the second-order partial derivatives of a function.

2. What is the Laplace operator?

The Laplace operator is a differential operator that is used to describe the behavior of a function in space. It is represented by the symbol ∇^2 and is defined as the sum of the second-order partial derivatives of a function with respect to its independent variables. In the context of PDEs, the Laplace operator is often used to describe the diffusion of heat or the spread of electric potential.

3. What is Laplace's equation?

Laplace's equation is a specific type of PDE that involves the Laplace operator. It is a second-order linear PDE that describes the behavior of a scalar function in space. In other words, it describes how a scalar quantity changes with respect to its independent variables. Laplace's equation is often used to model various physical phenomena, such as heat transfer, fluid flow, and electrostatics.

4. How is Laplace's equation solved?

Laplace's equation can be solved using a variety of methods, including separation of variables, the method of characteristics, and Fourier series. The specific method used will depend on the boundary conditions and the type of PDE being solved. In general, solving Laplace's equation involves finding a function that satisfies the equation and the given boundary conditions.

5. What are the applications of Laplace's equation?

Laplace's equation has a wide range of applications in physics, engineering, and mathematics. It is often used to model diffusion processes, such as heat flow and fluid flow. It is also used in electrostatics to describe the behavior of electric potential. In mathematics, Laplace's equation is used in the study of harmonic functions and in potential theory.

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