# Question:How do I solve two PDEs together while they are not simple?

## Question:How do I solve two PDEs together while they are not simple?

Maple 17

Dear users,

In my attached file I have two PDES, (PDE1 and PDE2). PDE1 is a function of v(t) and w(x,t) and PDE2 is also a function of v(t) and w(x,t). I can solve PDE2 if I say v(t) is 1 for example and you can see the plot. But what if I put v(t) back in PDE2 and want to find v(t) and w(x,t) from PDE1 and PDE2 together?

Many Thanks,

Baharm31

Define PDE Euler-Bernoulli Beam

 > restart:

Parametrs of piezoelectric and cantilever beam

 > Ys := 70*10^9: # Young's Modulus structure
 > Yp := 11.1*10^10: # Young's Modulus pieazo
 > ha := -0.00125: # Position
 > hb := 0.001: # Position
 > hc := 0.0015: # Position
 > d31 := -180*10^(-12): # Piezoelectric constant
 > b := 0.01: #Width of the beam
 > tb := 0.002:
 > epsilon33 := 15.92*10^(-9):
 > hp :=0.00025: # Position
 > hpc := 0.00125: # Position
 > YI := b*(Ys*(hb^3- ha^3)+Yp*(hc^3-hb^3))/3: # Bending stiffness of the composit cross section
 > cs := 0.564: # The equivqlent coefficient of strain rate damping
 > ca := 0: # Viscous air damping coefficient
 > Ibeam := (b * tb^3 )/12: # The equivalent moment of inertia
 > m := 0.101: # Mass of the structure
 > upsilon := - Yp*d31*b*(hc^2-hb^2)/(2*hp): # Coupling term
 > lb := 0.57:# Length of the structure (Cantilever Beam)
 > lp := 0.05:# Length of the Piezoelectric
 > R:= 10000: # Shunted resistor

Electrical circuit equation

 > PDE1:=(epsilon33 * b*lp / hp) * diff(v(t), t) + (v(t)/R)+ int(d31*Yp*hpc*b* diff(w(x, t),\$(x, 2))*diff(w(x, t), t),x = 0..lp)=0;
 (1.1.1.1)
 >
 >

PDE Equation

 > fn := 3.8:# Direct Excitation frequency;
 > wb(x,t) := 0.01*sin(fn*2*Pi*t):#Direct Excitation;
 > plot(wb(x,t),t = 0 .. 0.25*Pi,labels = [t,wb], labeldirections = ["horizontal", "vertical"], labelfont = ["HELVETICA", 15], linestyle = [longdash], axesfont = ["HELVETICA", "ROMAN", 10], legendstyle = [font = ["HELVETICA", 10], location = right],color = black);
 >
 (1.2.1)
 >
 > PDE2 := YI*diff(w(x, t),\$(x, 4))+ cs*Ibeam*diff(w(x, t),\$(x, 4))*diff(w(x, t), t)+ ca* diff(w(x, t), t) + m * diff(w(x, t),\$(t, 2))+ upsilon*v(t)*(Dirac(1,x) -Dirac(1,x-lp) ) =-m*diff(wb(x, t),\$(t, 2))-ca*diff(wb(x, t), t);#PDE
 (1.2.2)
 > tmax := 0.3:
 > xmin := 0:
 > xmax := lb:
 > N := 20:#NUMBER OF NODE POINT
 > bc1 := dw(xmin, t) = 0:
 > bc2 := dw(xmax, t) = 0:
 > bc3 := w(xmin, t) = 0:
 > ic1 := wl(x, 0) = 0:

Maple's pdsolve command

 >
 > bcs := { w(x,0)=0 , D[2](w)(x,0)=0 , w(0, t) = rhs(bc1), D[1](w)(0, t)= rhs(bc1), D[1,1](w)(lb,t) = rhs(bc2), D[1,1,1](w)(lb,t) = rhs(bc2)}; # Boundary conditions for PDE2.
 (2.1)
 > PDES := pdsolve(PDE2, bcs, numeric, time = t, range = 0 .. xmax, indepvars = [x, t], spacestep = (1/1000)*xmax, timestep = (1/1000)*tmax);
 >
 (2.2)
 > PDES:-plot3d(t = 0 .. tmax, x = 0 .. xmax, axes = boxed, orientation = [-120, 40], shading = zhue, transparency = 0.3);