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Question: Complex solution?

Question:Complex solution?

kh2n 220 Maple
complex plot
February 16 2011
0



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restart:with(MultiSeries):
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Digits:=30:
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variphi:=0.38:R1:=0.0009:R2:=8.75:E1:=1:E2:=1:R4:=177.6:C:=1.96:k:=4:H:=7:Pec_i:=-120:Phi:=1:Q:=1.9:R:=20:
> 

A:=(Q*(R*S+1-S)*(H+1/(1-exp(Pec_i)))*exp(C*Pec_i*S/(k*(R*S+1-S)))-H)*(1-exp(Pec_i*(1-S)/(R*S+1-S)))=1;

(1)
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eqn2:=Student:-Calculus1:-Roots(subs(H=7,R=20,C=1.96,k=4,Pec_i=-120,Q=1.9,A),S);

(2)
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S0:=.3592598898850950896270594340253413222287;

(3)
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F:=1/(R*S0+1-S0);

(4)
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w_0:=1/k*Pec_i*F*C;

(5)
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w_1:=Pec_i*F;

(6)
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M_1:=evalf(Q*Pec_i/k*(H+1/(1-exp(Pec_i))));

(7)
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M_2:=M_1*exp(w_0*S0);

(8)
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M_3:=w_1/(1-exp(w_1*(S0-1)));

(9)
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M_4:=M_3*exp(w_1*(S0-1));

(10)
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R3:=-38.16;

(11)
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A2:=variphi*sigma*Phi = -2*(((1/4*(cosh(l)-sinh(l))*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*w_0*(-l*w_0+E1*sigma)*(R1-1)*C^2*(l*C*w_1-sigma*E2*R1*k)*(l*w_0+E1*sigma)*M_3*R2*(-1/2*w_1+l)*exp(S0*(w_1-l)-w_1+2*l)+(1/4*(1/2*w_1+l)*(cosh(l)-sinh(l))*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*w_0*(-l*w_0+E1*sigma)*(R1-1)*C^2*(l*w_0+E1*sigma)*M_3*R2*exp(S0*(w_1+l)-w_1)+(-1/2*(l^2-1/2*l*w_0+E1*sigma+1/2*w_0^2)*(-l*w_0+E1*sigma)*(R1-1)*(l*R3*R2*cosh(l*(-1+S0))*R1+variphi*sinh(l*(-1+S0))*sigma)*C*M_1*k*exp(-(-w_0+l)*S0)+((-1/2*l*(R1-1)*C*R3*M_1*(l^2+1/2*l*w_0+E1*sigma+1/2*w_0^2)*k*exp((w_0+l)*S0)+(-l*w_0+E1*sigma)*(-1/2*H*k*R3*l*w_0*(R1-1)*cosh(l*S0)+(M_2*k*sigma*E1+1/2*M_2*w_0^2*k+l^2*M_2*k+1/4*w_1*M_4*w_0)*C*sinh(l*S0)))*R1*R2*cosh(l*(-1+S0))-(1/2*(R1-1)*sigma*variphi*C*M_1*(l^2+1/2*l*w_0+E1*sigma+1/2*w_0^2)*k*exp((w_0+l)*S0)+((M_2*k*sigma*E1+1/2*M_2*w_0^2*k+l^2*M_2*k+1/4*w_1*M_4*w_0)*C+1/2*H*k*sigma*variphi*w_0*(R1-1))*(-l*w_0+E1*sigma)*cosh(l*S0))*sinh(l*(-1+S0)))*(l*w_0+E1*sigma))*(l*C*w_1-sigma*E2*R1*k))*(l*C*w_1+sigma*E2*R1*k))*sinh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))-1/2*(w_0^2+4*E1*sigma+4*l^2)^(1/2)*((1/2*(cosh(l)-sinh(l))*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*(-l*w_0+E1*sigma)*(R1-1)*C^2*(l*C*w_1-sigma*E2*R1*k)*(l*w_0+E1*sigma)*M_3*R2*(-1/2*w_1+l)*exp(S0*(w_1-l)-w_1+2*l)+(l*C*w_1+sigma*E2*R1*k)*(1/2*(1/2*w_1+l)*(cosh(l)-sinh(l))*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*(-l*w_0+E1*sigma)*(R1-1)*C^2*(l*w_0+E1*sigma)*M_3*R2*exp(S0*(w_1+l)-w_1)+(1/2*C*k*M_1*(-w_0+l)*(R1-1)*(l*R3*R2*cosh(l*(-1+S0))*R1+variphi*sinh(l*(-1+S0))*sigma)*(-l*w_0+E1*sigma)*exp(-(-w_0+l)*S0)+((-1/2*C*k*R3*M_1*l*(w_0+l)*(R1-1)*exp((w_0+l)*S0)+(-l*w_0+E1*sigma)*(-H*k*R3*l*(R1-1)*cosh(l*S0)+C*(M_2*k*w_0+1/2*M_4*w_1)*sinh(l*S0)))*R1*R2*cosh(l*(-1+S0))-sinh(l*(-1+S0))*(1/2*C*k*M_1*sigma*variphi*(w_0+l)*(R1-1)*exp((w_0+l)*S0)+(-l*w_0+E1*sigma)*cosh(l*S0)*((M_2*k*w_0+1/2*M_4*w_1)*C+H*k*sigma*variphi*(R1-1))))*(l*w_0+E1*sigma))*(l*C*w_1-sigma*E2*R1*k)))*cosh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))+C*k*M_1*w_0*exp(1/2*S0*w_0)*(R1-1)*(l*C*w_1-sigma*E2*R1*k)*(l*C*w_1+sigma*E2*R1*k)*(l*R3*R2*cosh(l*(-1+S0))*R1+variphi*sinh(l*(-1+S0))*sigma)*(E1*sigma+l^2)))*sinh(1/2*(-1+S0)*((w_1^2+4*l^2)*C^2+4*C*sigma*E2*R1*k)^(1/2)/C)-1/8*(-cosh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))*(w_0^2+4*E1*sigma+4*l^2)^(1/2)+sinh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))*w_0)*((w_1^2+4*l^2)*C^2+4*C*sigma*E2*R1*k)^(1/2)*(-l*w_0+E1*sigma)*(l*w_0+E1*sigma)*((-C*M_3*R2*(cosh(l)-sinh(l))*(R1-1)*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*(l*C*w_1-sigma*E2*R1*k)*exp(S0*(w_1-l)-w_1+2*l)+(C*M_3*R2*(cosh(l)-sinh(l))*(R1-1)*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*exp(S0*(w_1+l)-w_1)+2*M_4*(R1*R2*cosh(l*(-1+S0))*sinh(l*S0)-cosh(l*S0)*sinh(l*(-1+S0)))*(l*C*w_1-sigma*E2*R1*k))*(l*C*w_1+sigma*E2*R1*k))*cosh(1/2*(-1+S0)*((w_1^2+4*l^2)*C^2+4*C*sigma*E2*R1*k)^(1/2)/C)-2*C*k*E2*M_3*R1*sigma*R2*exp(1/2*w_1*(-1+S0)+l)*(cosh(l)-sinh(l))*(R1-1)*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)))*Phi/H/k/(l^2*C^2*w_1^2-sigma^2*E2^2*R1^2*k^2)/sinh(1/2*(-1+S0)*((w_1^2+4*l^2)*C^2+4*C*sigma*E2*R1*k)^(1/2)/C)/(R1*R2*cosh(l*(-1+S0))*sinh(l*S0)-cosh(l*S0)*sinh(l*(-1+S0)))/(-cosh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))*(w_0^2+4*E1*sigma+4*l^2)^(1/2)+sinh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))*w_0)/(-l^2*w_0^2+E1^2*sigma^2):
> 

plots:-animate(plot,[[rhs,lhs](A2),sigma=-350..5,-130..10],l=1.5..3.5,frames=20);

 
> 

plots:-animate(plot,[[rhs,lhs](A2),sigma=-350..5,-130..10],l=0.00001..4.5,frames=20);

 

  

From The above two plots we can see that equation A2 has multi solutions in the interval (3.5, 20] and the realistic range of the

solution falls in the range sigma=[-25, 0]. But for the interval (0, 3.5] the equation A2 has solutions but those solution shows a

big Jump and also the range is not realistic the range for l=(0, 3.5] is sigma>-300. Now I am trying to find the complex solution

of equation A2 for the specific interval  l=(0, 3.5].

Here I have tried to find the complex solution for some values of l. But there are still multi

complex solutions for each values of l in the interval (0,3.5]. Is there any way to plot the

complex solution of equation A2 in the specific domain (0,3.5]. Then from which we can make

that which of the complex solution for the specific value of l is in the right range.

> 

fsolve( [subs(l=0.5,A2)],sigma=-29.3343821397092591432961519014-35.8670658288521588089836207749*I, complex);

(12)
> 

fsolve( [subs(l=0.5,A2)],sigma=-29.3343821397092591432961519014+35.8670658288521588089836207749*I, complex);

(13)
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fsolve( [subs(l=1.1,A2)],sigma=-30.5394496972056264187225441025-25.2641666423804182852890108017*I, complex);

(14)
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fsolve( [subs(l=1.1,A2)],sigma=-30.5394496972056264187225441025+25.2641666423804182852890108017*I, complex);

(15)
> 

fsolve( [subs(l=1.2,A2)],sigma=-29.4430686031432971298788061327+34.8802142452845002850519148160*I, complex);

(16)
> 

fsolve( [subs(l=1.3,A2)],sigma=-29.5045289777383571301351085832+34.3262443345923597082905511940*I, complex);

(17)
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fsolve( [subs(l=1.4,A2)],sigma=-29.5708589774295808475971897770+33.7315397902111513718611413301*I, complex);

(18)
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fsolve( [subs(l=1.5,A2)],sigma=-150, complex);

(19)
> 

fsolve( [subs(l=2,A2)],sigma=-100, complex);

(20)
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fsolve( [subs(l=2.5,A2)],sigma=-29.9804855039278517421286313336-30.1196432492497560051550511400*I, complex);

(21)
> 

fsolve( [subs(l=2.6,A2)],sigma=-29.9804855039278517421286313336-30.1196432492497560051550511400*I, complex);

(22)
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fsolve( [subs(l=2.7,A2)],sigma=-29.9804855039278517421286313336-30.1196432492497560051550511400*I, complex);

(23)
> 

fsolve( [subs(l=2.8,A2)],sigma=-29.9804855039278517421286313336-30.1196432492497560051550511400*I, complex);

(24)
> 

for K from 0.5 by 0.1  to 3.5 do
fsolve(subs(l=K, A2),sigma=-29.5708589774295808475971897770+33.7315397902111513718611413301*I..-30.9618204794783814736049777007-21.5088815441933078811796364582*I):                                                                                              if not has(%,fsolve) then lprint(K,%) end if
od:

Error, (in fsolve) -29.5708589774295808475971897770+33.7315397902111513718611413301*I .. -30.9618204794783814736049777007-21.5088815441933078811796364582*I is an invalid range

  
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