Explore BrainMass
Share

Partial Differential Equations

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

1. Consider the first order PDE,

∂u/∂t = ct(∂u/∂x) -&#8734; < x < &#8734; c does not equal 0

a) Find the fundamental solution

b) Use the fundamental solution and convolution to find a formula for the solution to:

∂u/∂t = ct(∂u/∂x) -&#8734; < x < &#8734; c does not equal 0 u(x,0) = f(x)

c) Use the method of characteristics to find the solution to part b.(Make sure your answers agree)

© BrainMass Inc. brainmass.com October 24, 2018, 11:05 pm ad1c9bdddf
https://brainmass.com/math/partial-differential-equations/partial-differential-equations-functions-180856

Attachments

Solution Preview

Hello

Please find the solution in the attached file.

1. Consider the first order PDE,

∂u/∂t = ct(∂u/∂x) -∞ < x < ∞ c does not equal 0

a) Find the fundamental solution

b) Use the fundamental solution and convolution to find a formula for ...

Solution Summary

This finds the fundamental solution of a first-order partial differential equation, and uses that solution and convolution, as well as method of characteristics, in another problem. The expert uses the method of characteristics to find the function.

$2.19
See Also This Related BrainMass Solution

Partial Differential Equations : Heat Equations

1) Let A(x,y) be the area of a rectangle not degenerated of dimensions x and y, in a way that the rectangle is inside a circle of a radius of 10. Determine the domain and the range of this function.

2) The wave equation (c^2 &#8706;^2 u / &#8706; x^2 = &#8706;^2 u / &#8706; t^2) and the heat equation (c &#8706;^2 u / &#8706; x^2 = &#8706; u / &#8706; t) are two of the most important equations of physics (c is a constant). They are called partial differential equations. Show the following:

a) u = cos x cos ct and u = e^x cosh ct satisfies the wave equation.

b) u = e^-ct sin x and u = t^-1/2 e^[(-x^2)/(4ct)] satisfies the heat equation.

View Full Posting Details