PDE6.5

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Transform the BVP into one with homogeneous boundary conditions.

u_{t}=u_{{xx}}+w(x,t)\,u_{x}(0,t)=\alpha (t)\,
u_{x}(1,t)=\beta (t)\,
u(x,0)=\phi (x)\,

0<x<1\,
t>0\,



Let v(x,t)=u(x,t)-{\frac  {\beta (t)}{2}}x^{2}-(x-{\frac  {x^{2}}{2}})\alpha (t)\,

v_{x}=u_{x}-\beta (t)x-\alpha (t)+\alpha (t)x\,

v_{{xx}}=u_{{xx}}-\beta (t)+\alpha (t)\,

v_{t}=u_{t}-{\frac  {\beta '(t)}{2}}x^{2}-(x-{\frac  {x^{2}}{2}})\alpha '(t)\,

Now the new initial boundary value problem can be stated.

The new DE is v_{t}=v_{{xx}}+\beta (t)-\alpha (t)-{\frac  {\beta '(t)}{2}}x^{2}-(x-{\frac  {x^{2}}{2}})\alpha '(t)+w(x,t)\,

The new IC is v(x,0)=u(x,0)-{\frac  {\beta (0)}{2}}x^{2}-(x-{\frac  {x^{2}}{2}})\alpha (0)\,

The new BCs are

v_{x}(0,t)=u_{x}(0,t)-\alpha (t)=0\,

v_{x}(1,t)=u_{x}(1,t)-\beta (t)=0\,

Partial Differential Equations

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