Explore BrainMass
Share

Explore BrainMass

    Partial differential equation

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

    PDE Utt = Uxx+sin(3&#61552;x) 0<x<1 0<t<&#61605;

    BC's U(0,t)=0
    U(1,t)= 0

    IC's U(x,0)= 2sin(2&#61552;x)
    Ut(x,0)= 0

    © BrainMass Inc. brainmass.com October 9, 2019, 7:44 pm ad1c9bdddf
    https://brainmass.com/math/partial-differential-equations/partial-differential-equation-127759

    Attachments

    Solution Preview

    Please see the attached file.

    PDE Utt = Uxx+sin(3x) 0<x<1 0<t<

    BC's U(0,t)=0
    U(1,t)= 0

    IC's U(x,0)= 2sin(2x)
    Ut(x,0)= 0

    We first must consider the homogenous equation:

    We start as always with separating the wave equation:

    Plugging it back into the homogenous equation we obtain:

    Each side is independent of the other (since each side is a function of a single independent variable).

    This can occur if an only if both sides equal the same constant number.

    Hence:

    The spatial equation becomes:

    Case 1:

    The general solution of the equation is:

    Using boundary conditions we have:

    Hence for we get only the trivial solution.

    Case 2:

    The general solution of the equation is:

    Using boundary conditions we have:

    Hence for we get only the trivial solution.

    So we are left with case 3:

    The general solution of the equation is:

    ...

    Solution Summary

    This provides an example of solving a partial differential equation. The functions of partial differential equations are examined.

    $2.19