Linearity

squenshl

I'm trying to see if [tex]\rho[/tex]u_tt + EIu_xxxx = 0 is linear or non-linear where [tex]\rho[/tex], E and I are constants.

I got L(u+v) = [tex]\rho[/tex][tex]\delta[/tex]²u²/[tex]\delta[/tex]t² + EI[tex]\delta[/tex]⁴u²/[tex]\delta[/tex]x⁴ + [tex]\rho[/tex][tex]\delta[/tex]²uv/[tex]\delta[/tex]t² + EI[tex]\delta[/tex]⁴uv/[tex]\delta[/tex]x⁴ = Lu + Lv. Does this mean it's linear or is there more to do.

arildno

That is enough.

squenshl

Cheers.
What about this one then.
u_t - [tex]\alpha^2[/tex][tex]\nabla^2[/tex]u = ru(M -u) where [tex]\alpha[/tex], r & M are constants.

u_t - [tex]\alpha^2[/tex][tex]\nabla^2[/tex]u - ru(M -u) = 0
L(u+v+w) = u_t(u+v+w) + [tex]\alpha^2[/tex][tex]\nabla^2[/tex]u(u+v+w) - ru(M-u)(u+v+w) = u_tt + [tex]\alpha^2[/tex][tex]\nabla^2[/tex]u² - ru²(M-u) + u_tv + [tex]\alpha^2[/tex][tex]\nabla^2[/tex]uv - ruv(M-u) + u_tw + [tex]\alpha^2[/tex][tex]\nabla^2[/tex]uw - ruw(M-u) = Lu + Lv + Lw

squenshl

Are these equation linear or non-linear?

u_t + (1-u)u_x = 0
u_xx + e^xu_tt = sin(x)
u_xx + u_xy + u_yy + u_x = t²

squenshl

Someone help please.