Math Solutions

(Steps Shown) We have (partial ^2u)/(∂ x^2)+(partial ^2u)/(∂ y^2)=(1)/(c^2)(partial ^2u)/(partial t^2), xin (0,a), yin (0,b), t>0 , B.C. :u(0,y,t)=u(a,y,t)=u(x,0,t)=u(x,b,t)=0

Question: We have

\[\begin{aligned} & \frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}u}{\partial {{y}^{2}}}=\frac{1}{{{c}^{2}}}\frac{{{\partial }^{2}}u}{\partial {{t}^{2}}},\quad x\in (0,a),\quad y\in (0,b),\quad t>0 \\ & \text{B}\text{.C}\text{. }:u(0,y,t)=u(a,y,t)=u(x,0,t)=u(x,b,t)=0 \\ \end{aligned}\]

Show that the solution can be expressed as

\[\begin{aligned} & u(x,y,t)=\sum\limits_{m=1}^{\infty }{\sum\limits_{n=1}^{\infty }{\left\{ {{A}_{mn}}\cdot \cos \sqrt{\frac{{{c}^{2}}{{n}^{2}}{{\pi }^{2}}}{{{b}^{2}}}+\frac{{{c}^{2}}{{m}^{2}}{{\pi }^{2}}}{{{a}^{2}}}}+{{B}_{mn}}\cdot \sin \sqrt{\frac{{{c}^{2}}{{n}^{2}}{{\pi }^{2}}}{{{b}^{2}}}+\frac{{{c}^{2}}{{m}^{2}}{{\pi }^{2}}}{{{a}^{2}}}} \right\}}} \\ & \,\,\,\,\,\,\cdot \sin \frac{rn\pi }{a}x\cdot \sin \frac{n\pi }{b}y \\ \end{aligned}\]

Price: $2.99
Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document


∴ Other downloads you may be interested in ∴

log in to your account

Forgot password?
Don't have a membership account?
REGISTER

reset password

Back to
log in

sign up

Back to
log in