NIMA112

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These are questions asked by NIMA112

hello,

I computed some algebraic calculation, but near point -1 there is some issue like disconvergencies. Can anyone help in this regard? In the final graph, as seen the paths of the curves are close together and convergence does not occur AROUND POINT -1.

restart

Digite := 30

30

(1)

beta := 2.5; lambda := 0.1e-1; b := Pi; a := Pi; alpha := 0; y[1] := 1.5; y[2] := 1.5; x[1] := -1; x[2] := 1; Q[1] := 40; Q[2] := 35; T[1] := 20

2.5

 

0.1e-1

 

Pi

 

Pi

 

0

 

1.5

 

1.5

 

-1

 

1

 

40

 

35

(2)

v := (2*n-1)*Pi/(2*b)

Delta := exp(2*v*a)*(alpha*v+beta)*(1+lambda)-(1-lambda)*(alpha*v-beta)

omega := Pi/(2*b)

P[1] := ((1+lambda)*exp(-v*abs(x-xi))+(1-lambda)*exp(v*(x+xi)))*exp(2*v*a)+(1+lambda)*exp(-v*(x+xi))+(1-lambda)*exp(v*abs(x-xi))

P[2] := ((1+lambda)*exp(-v*abs(x-xi))+(1-lambda)*exp(v*(x+xi)))*exp(2*v*a)-(1+lambda)*exp(-v*(x+xi))-(1-lambda)*exp(v*abs(x-xi))

P[3] := P[1]*(-alpha^2*v-alpha*beta+alpha*v)+beta*P[2]

G[11] := (sum((alpha*P[1]*(1-lambda)*(alpha*v-beta)*exp(-2*v*a)+(1+lambda)*P[3])*(cos(v*(y-eta))-cos(v*(y+eta)))/(v*(exp(2*v*a)*(alpha*v+beta)*(1+lambda)-(1-lambda)*(alpha*v-beta))), n = 1 .. 80))/(2*b*(1+lambda))+(2*(1+lambda)*alpha*b/Pi*.25)*ln((1+2*exp(-omega*abs(x-xi))*cos(omega*(y-eta))+exp(-2*omega*abs(x-xi)))*(1-2*exp(-omega*abs(x-xi))*cos(omega*(y+eta))+exp(-2*omega*abs(x-xi)))*(1+2*exp(-omega*(2*a+x+xi))*cos(omega*(y-eta))+exp(-2*omega*(2*a+x+xi)))*(1-2*exp(-omega*(2*a+x+xi))*cos(omega*(y+eta))+exp(-2*omega*(2*a+x+xi)))/((1-2*exp(-omega*abs(x-xi))*cos(omega*(y-eta))+exp(-2*omega*abs(x-xi)))*(1+2*exp(-omega*abs(x-xi))*cos(omega*(y+eta))+exp(-2*omega*abs(x-xi)))*(1-2*exp(-omega*(2*a+x+xi))*cos(omega*(y-eta))+exp(-2*omega*(2*a+x+xi)))*(1+2*exp(-omega*(2*a+x+xi))*cos(omega*(y+eta))+exp(-2*omega*(2*a+x+xi)))))/(2*b*(1+lambda))+(2*(1-lambda)*alpha*b/Pi*.25)*ln((1+2*exp(omega*(x+xi))*cos(omega*(y-eta))+exp(2*omega*(x+xi)))*(1-2*exp(omega*(x+xi))*cos(omega*(y+eta))+exp(2*omega*(x+xi)))*(1+2*exp(-omega*(2*a-abs(x-xi)))*cos(omega*(y-eta))+exp(-2*omega*(2*a-abs(x-xi))))*(1-2*exp(-omega*(2*a-abs(x-xi)))*cos(omega*(y+eta))+exp(-2*omega*(2*a-abs(x-xi))))/((1-2*exp(omega*(x+xi))*cos(omega*(y-eta))+exp(2*omega*(x+xi)))*(1+2*exp(omega*(x+xi))*cos(omega*(y+eta))+exp(2*omega*(x+xi)))*(1-2*exp(-omega*(2*a-abs(x-xi)))*cos(omega*(y-eta))+exp(-2*omega*(2*a-abs(x-xi))))*(1+2*exp(-omega*(2*a-abs(x-xi)))*cos(omega*(y+eta))+exp(-2*omega*(2*a-abs(x-xi))))))/(2*b*(1+lambda))

g[12] := lambda*((alpha*v+beta)*exp(v*(2*a+x))+(alpha*v-beta)*exp(-v*x))*exp(-v*xi)/(v*Delta)

G[12] := (sum(g[12]*(cos(v*(y-eta))-cos(v*(y+eta))), n = 1 .. 80))/b

phi[1] := int(int(G[11]*Q[1]*Dirac(xi-x[1])*Dirac(eta-y[1]), xi = -a .. 0), eta = 0 .. b)+int(int(G[12]*Q[2]*Dirac(xi-x[2])*Dirac(eta-y[2]), xi = 0 .. infinity), eta = 0 .. b)

Z[1] := diff(phi[1], x)

psi[1] := int(Z[1], y)

plot3d(psi[1], x = -a .. 0, y = 0 .. b)

 

with(plots)

contourplot(psi[1], x = -a .. 0, y = 0 .. b)

 

 

Download sai_1.mw

Hi everyone.

Could you please help me to obtain the results by 'solve'?

Is there any way such as numerical methods in this regard?

Fung.mws

How I can ontain results without Rootof !!

Thanks

help.mw

restart

EQ1 := c[11]*exp(-n*Pi*a/b)+c[12]*exp(n*Pi*a/b) = 0

c[11]*exp(-n*Pi*a/b)+c[12]*exp(n*Pi*a/b) = 0

(1)

EQ2 := c[11]*n*Pi/b-c[12]*n*Pi/b+b^2*Q[0, 1]*sin(n*Pi*y[0, 1]/b)*(n*Pi*exp(-n*Pi*x[0, 1]/b)/b+n*Pi*exp(n*Pi*x[0, 1]/b)/b)/(n*Pi) = c[21]*n*Pi/b-c[22]*n*Pi/b

c[11]*n*Pi/b-c[12]*n*Pi/b+b^2*Q[0, 1]*sin(n*Pi*y[0, 1]/b)*(n*Pi*exp(-n*Pi*x[0, 1]/b)/b+n*Pi*exp(n*Pi*x[0, 1]/b)/b)/(n*Pi) = c[21]*n*Pi/b-c[22]*n*Pi/b

(2)

c[21] := -b^2*Q[0, 2]*sin(n*Pi*y[0, 2]/b)*exp(-n*Pi*x[0, 2]/b)/(n*Pi)

-b^2*Q[0, 2]*sin(n*Pi*y[0, 2]/b)*exp(-n*Pi*x[0, 2]/b)/(n*Pi)

(3)

EQ3 := c[11]+c[12]+b^2*Q[0, 1]*sin(n*Pi*y[0, 1]/b)*(exp(-n*Pi*x[0, 1]/b)-exp(n*Pi*x[0, 1]/b))/(n*Pi) = lambda.(c[21]+c[22])

c[11]+c[12]+b^2*Q[0, 1]*sin(n*Pi*y[0, 1]/b)*(exp(-n*Pi*x[0, 1]/b)-exp(n*Pi*x[0, 1]/b))/(n*Pi) = lambda.(-b^2*Q[0, 2]*sin(n*Pi*y[0, 2]/b)*exp(-n*Pi*x[0, 2]/b)/(n*Pi)+c[22])

(4)

solve({EQ1, EQ2, EQ3}, {c[11], c[12], c[22]})

{c[11] = RootOf(exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]-exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]-_Z*n*Pi*exp(-n*Pi*a/b)+_Z*n*Pi*exp(n*Pi*a/b)+(lambda.((exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+2*b^2*Q[0, 2]*sin(n*Pi*y[0, 2]/b)*exp(-n*Pi*x[0, 2]/b)*exp(n*Pi*a/b)+_Z*n*Pi*exp(-n*Pi*a/b)+_Z*n*Pi*exp(n*Pi*a/b))/(n*exp(n*Pi*a/b))))*n*exp(n*Pi*a/b)), c[12] = -RootOf(exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]-exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]-_Z*n*Pi*exp(-n*Pi*a/b)+_Z*n*Pi*exp(n*Pi*a/b)+(lambda.((exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+2*b^2*Q[0, 2]*sin(n*Pi*y[0, 2]/b)*exp(-n*Pi*x[0, 2]/b)*exp(n*Pi*a/b)+_Z*n*Pi*exp(-n*Pi*a/b)+_Z*n*Pi*exp(n*Pi*a/b))/(n*exp(n*Pi*a/b))))*n*exp(n*Pi*a/b))*exp(-n*Pi*a/b)/exp(n*Pi*a/b), c[22] = -(exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+b^2*Q[0, 2]*sin(n*Pi*y[0, 2]/b)*exp(-n*Pi*x[0, 2]/b)*exp(n*Pi*a/b)+RootOf(exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]-exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]-_Z*n*Pi*exp(-n*Pi*a/b)+_Z*n*Pi*exp(n*Pi*a/b)+(lambda.((exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+2*b^2*Q[0, 2]*sin(n*Pi*y[0, 2]/b)*exp(-n*Pi*x[0, 2]/b)*exp(n*Pi*a/b)+_Z*n*Pi*exp(-n*Pi*a/b)+_Z*n*Pi*exp(n*Pi*a/b))/(n*exp(n*Pi*a/b))))*n*exp(n*Pi*a/b))*Pi*n*exp(-n*Pi*a/b)+RootOf(exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]-exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]-_Z*n*Pi*exp(-n*Pi*a/b)+_Z*n*Pi*exp(n*Pi*a/b)+(lambda.((exp(-n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+exp(n*Pi*x[0, 1]/b)*exp(n*Pi*a/b)*b^2*sin(n*Pi*y[0, 1]/b)*Q[0, 1]+2*b^2*Q[0, 2]*sin(n*Pi*y[0, 2]/b)*exp(-n*Pi*x[0, 2]/b)*exp(n*Pi*a/b)+_Z*n*Pi*exp(-n*Pi*a/b)+_Z*n*Pi*exp(n*Pi*a/b))/(n*exp(n*Pi*a/b))))*n*exp(n*Pi*a/b))*Pi*n*exp(n*Pi*a/b))/(Pi*n*exp(n*Pi*a/b))}

(5)

``

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