AHSAN

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These are replies submitted by AHSAN

@tomleslie  Thanks sir

@acer still could not find the solution, can you please make correction

 


 

restart

"u(y):=-(3 beta ((2 p^3 sigma^4)/3+((2 p^3 y^2)/3-(4 q)/3) sigma^2-(4 p^2 y (k+1) sigma)/3+p (k+1)^2) (y+sigma) (y-sigma))/(8 sigma^2)+(-p sigma^3+(p y^2-k+1) sigma-(k+1) y)/(2 sigma)"

proc (y) options operator, arrow, function_assign; -(3/8)*beta*((2/3)*p^3*sigma^4+((2/3)*p^3*y^2-(4/3)*q)*sigma^2-(4/3)*p^2*y*(k+1)*sigma+p*(k+1)^2)*(y+sigma)*(y-sigma)/sigma^2+(1/2)*(-p*sigma^3+(p*y^2-k+1)*sigma-(k+1)*y)/sigma end proc

(1)

ode := diff(theta(y), y, y)+G*(diff(u(y), y))^2+G*beta*(diff(u(y), y))^4

diff(diff(theta(y), y), y)+G*(-(3/8)*beta*((4/3)*p^3*y*sigma^2-(4/3)*p^2*(k+1)*sigma)*(y+sigma)*(y-sigma)/sigma^2-(3/8)*beta*((2/3)*p^3*sigma^4+((2/3)*p^3*y^2-(4/3)*q)*sigma^2-(4/3)*p^2*y*(k+1)*sigma+p*(k+1)^2)*(y-sigma)/sigma^2-(3/8)*beta*((2/3)*p^3*sigma^4+((2/3)*p^3*y^2-(4/3)*q)*sigma^2-(4/3)*p^2*y*(k+1)*sigma+p*(k+1)^2)*(y+sigma)/sigma^2+(1/2)*(2*p*sigma*y-k-1)/sigma)^2+G*beta*(-(3/8)*beta*((4/3)*p^3*y*sigma^2-(4/3)*p^2*(k+1)*sigma)*(y+sigma)*(y-sigma)/sigma^2-(3/8)*beta*((2/3)*p^3*sigma^4+((2/3)*p^3*y^2-(4/3)*q)*sigma^2-(4/3)*p^2*y*(k+1)*sigma+p*(k+1)^2)*(y-sigma)/sigma^2-(3/8)*beta*((2/3)*p^3*sigma^4+((2/3)*p^3*y^2-(4/3)*q)*sigma^2-(4/3)*p^2*y*(k+1)*sigma+p*(k+1)^2)*(y+sigma)/sigma^2+(1/2)*(2*p*sigma*y-k-1)/sigma)^4

(2)

dsolve(ode)

theta(y) = Int(Int(-(1/256)*G*(16*beta^3*p^6*sigma^4*y^6+16*beta^3*k*p^5*sigma^5*y^3-48*beta^3*k*p^5*sigma^3*y^5+16*beta^3*p^5*sigma^5*y^3-48*beta^3*p^5*sigma^3*y^5+4*beta^3*k^2*p^4*sigma^6-24*beta^3*k^2*p^4*sigma^4*y^2+60*beta^3*k^2*p^4*sigma^2*y^4-32*beta^3*p^3*q*sigma^4*y^4+8*beta^3*k*p^4*sigma^6-48*beta^3*k*p^4*sigma^4*y^2+120*beta^3*k*p^4*sigma^2*y^4-32*beta^2*p^4*sigma^4*y^4+12*beta^3*k^3*p^3*sigma^3*y-36*beta^3*k^3*p^3*sigma*y^3-16*beta^3*k*p^2*q*sigma^5*y+48*beta^3*k*p^2*q*sigma^3*y^3+4*beta^3*p^4*sigma^6-24*beta^3*p^4*sigma^4*y^2+60*beta^3*p^4*sigma^2*y^4+36*beta^3*k^2*p^3*sigma^3*y-108*beta^3*k^2*p^3*sigma*y^3-16*beta^3*p^2*q*sigma^5*y+48*beta^3*p^2*q*sigma^3*y^3-16*beta^2*k*p^3*sigma^5*y+64*beta^2*k*p^3*sigma^3*y^3+9*beta^3*k^4*p^2*y^2-24*beta^3*k^2*p*q*sigma^2*y^2+36*beta^3*k*p^3*sigma^3*y-108*beta^3*k*p^3*sigma*y^3+16*beta^3*q^2*sigma^4*y^2-16*beta^2*p^3*sigma^5*y+64*beta^2*p^3*sigma^3*y^3+36*beta^3*k^3*p^2*y^2-48*beta^3*k*p*q*sigma^2*y^2+12*beta^3*p^3*sigma^3*y-36*beta^3*p^3*sigma*y^3+8*beta^2*k^2*p^2*sigma^4-48*beta^2*k^2*p^2*sigma^2*y^2+32*beta^2*p*q*sigma^4*y^2+54*beta^3*k^2*p^2*y^2-24*beta^3*p*q*sigma^2*y^2+16*beta^2*k*p^2*sigma^4-96*beta^2*k*p^2*sigma^2*y^2+16*beta*p^2*sigma^4*y^2+36*beta^3*k*p^2*y^2+12*beta^2*k^3*p*sigma*y-16*beta^2*k*q*sigma^3*y+8*beta^2*p^2*sigma^4-48*beta^2*p^2*sigma^2*y^2+9*beta^3*p^2*y^2+36*beta^2*k^2*p*sigma*y-16*beta^2*q*sigma^3*y-16*beta*k*p*sigma^3*y+36*beta^2*k*p*sigma*y-16*beta*p*sigma^3*y+12*beta^2*p*sigma*y+4*beta*k^2*sigma^2+8*beta*k*sigma^2+16*sigma^4+4*beta*sigma^2)*(4*beta*p^3*sigma^2*y^3+2*beta*k*p^2*sigma^3-6*beta*k*p^2*sigma*y^2+2*beta*p^2*sigma^3-6*beta*p^2*sigma*y^2+3*beta*k^2*p*y-4*beta*q*sigma^2*y+6*beta*k*p*y-4*p*sigma^2*y+3*beta*p*y+2*k*sigma+2*sigma)^2/sigma^8, y), y)+_C1*y+_C2

(3)

bc := theta(-sigma) = 0, theta(sigma) = 1

theta(-sigma) = 0, theta(sigma) = 1

(4)

sol := dsolve({bc, ode})

theta(y) = Int(Int(-(1/256)*G*(16*_z1^6*beta^3*p^6*sigma^4-48*_z1^5*beta^3*k*p^5*sigma^3+16*_z1^3*beta^3*k*p^5*sigma^5-48*_z1^5*beta^3*p^5*sigma^3+16*_z1^3*beta^3*p^5*sigma^5+60*_z1^4*beta^3*k^2*p^4*sigma^2-32*_z1^4*beta^3*p^3*q*sigma^4-24*_z1^2*beta^3*k^2*p^4*sigma^4+4*beta^3*k^2*p^4*sigma^6+120*_z1^4*beta^3*k*p^4*sigma^2-32*_z1^4*beta^2*p^4*sigma^4-48*_z1^2*beta^3*k*p^4*sigma^4+8*beta^3*k*p^4*sigma^6+60*_z1^4*beta^3*p^4*sigma^2-36*_z1^3*beta^3*k^3*p^3*sigma+48*_z1^3*beta^3*k*p^2*q*sigma^3-24*_z1^2*beta^3*p^4*sigma^4+12*_z1*beta^3*k^3*p^3*sigma^3-16*_z1*beta^3*k*p^2*q*sigma^5+4*beta^3*p^4*sigma^6-108*_z1^3*beta^3*k^2*p^3*sigma+48*_z1^3*beta^3*p^2*q*sigma^3+64*_z1^3*beta^2*k*p^3*sigma^3+36*_z1*beta^3*k^2*p^3*sigma^3-16*_z1*beta^3*p^2*q*sigma^5-16*_z1*beta^2*k*p^3*sigma^5-108*_z1^3*beta^3*k*p^3*sigma+64*_z1^3*beta^2*p^3*sigma^3+9*_z1^2*beta^3*k^4*p^2-24*_z1^2*beta^3*k^2*p*q*sigma^2+16*_z1^2*beta^3*q^2*sigma^4+36*_z1*beta^3*k*p^3*sigma^3-16*_z1*beta^2*p^3*sigma^5-36*_z1^3*beta^3*p^3*sigma+36*_z1^2*beta^3*k^3*p^2-48*_z1^2*beta^3*k*p*q*sigma^2-48*_z1^2*beta^2*k^2*p^2*sigma^2+32*_z1^2*beta^2*p*q*sigma^4+12*_z1*beta^3*p^3*sigma^3+8*beta^2*k^2*p^2*sigma^4+54*_z1^2*beta^3*k^2*p^2-24*_z1^2*beta^3*p*q*sigma^2-96*_z1^2*beta^2*k*p^2*sigma^2+16*_z1^2*beta*p^2*sigma^4+16*beta^2*k*p^2*sigma^4+36*_z1^2*beta^3*k*p^2-48*_z1^2*beta^2*p^2*sigma^2+12*_z1*beta^2*k^3*p*sigma-16*_z1*beta^2*k*q*sigma^3+8*beta^2*p^2*sigma^4+9*_z1^2*beta^3*p^2+36*_z1*beta^2*k^2*p*sigma-16*_z1*beta^2*q*sigma^3-16*_z1*beta*k*p*sigma^3+36*_z1*beta^2*k*p*sigma-16*_z1*beta*p*sigma^3+12*_z1*beta^2*p*sigma+4*beta*k^2*sigma^2+8*beta*k*sigma^2+16*sigma^4+4*beta*sigma^2)*(4*_z1^3*beta*p^3*sigma^2-6*_z1^2*beta*k*p^2*sigma+2*beta*k*p^2*sigma^3-6*_z1^2*beta*p^2*sigma+2*beta*p^2*sigma^3+3*_z1*beta*k^2*p-4*_z1*beta*q*sigma^2+6*_z1*beta*k*p-4*_z1*p*sigma^2+3*_z1*beta*p+2*k*sigma+2*sigma)^2/sigma^8, _z1 = sigma .. _z1), _z1 = sigma .. y)+(1/2)*(Int(Int(-(1/256)*G*(16*_z1^6*beta^3*p^6*sigma^4-48*_z1^5*beta^3*k*p^5*sigma^3+16*_z1^3*beta^3*k*p^5*sigma^5-48*_z1^5*beta^3*p^5*sigma^3+16*_z1^3*beta^3*p^5*sigma^5+60*_z1^4*beta^3*k^2*p^4*sigma^2-32*_z1^4*beta^3*p^3*q*sigma^4-24*_z1^2*beta^3*k^2*p^4*sigma^4+4*beta^3*k^2*p^4*sigma^6+120*_z1^4*beta^3*k*p^4*sigma^2-32*_z1^4*beta^2*p^4*sigma^4-48*_z1^2*beta^3*k*p^4*sigma^4+8*beta^3*k*p^4*sigma^6+60*_z1^4*beta^3*p^4*sigma^2-36*_z1^3*beta^3*k^3*p^3*sigma+48*_z1^3*beta^3*k*p^2*q*sigma^3-24*_z1^2*beta^3*p^4*sigma^4+12*_z1*beta^3*k^3*p^3*sigma^3-16*_z1*beta^3*k*p^2*q*sigma^5+4*beta^3*p^4*sigma^6-108*_z1^3*beta^3*k^2*p^3*sigma+48*_z1^3*beta^3*p^2*q*sigma^3+64*_z1^3*beta^2*k*p^3*sigma^3+36*_z1*beta^3*k^2*p^3*sigma^3-16*_z1*beta^3*p^2*q*sigma^5-16*_z1*beta^2*k*p^3*sigma^5-108*_z1^3*beta^3*k*p^3*sigma+64*_z1^3*beta^2*p^3*sigma^3+9*_z1^2*beta^3*k^4*p^2-24*_z1^2*beta^3*k^2*p*q*sigma^2+16*_z1^2*beta^3*q^2*sigma^4+36*_z1*beta^3*k*p^3*sigma^3-16*_z1*beta^2*p^3*sigma^5-36*_z1^3*beta^3*p^3*sigma+36*_z1^2*beta^3*k^3*p^2-48*_z1^2*beta^3*k*p*q*sigma^2-48*_z1^2*beta^2*k^2*p^2*sigma^2+32*_z1^2*beta^2*p*q*sigma^4+12*_z1*beta^3*p^3*sigma^3+8*beta^2*k^2*p^2*sigma^4+54*_z1^2*beta^3*k^2*p^2-24*_z1^2*beta^3*p*q*sigma^2-96*_z1^2*beta^2*k*p^2*sigma^2+16*_z1^2*beta*p^2*sigma^4+16*beta^2*k*p^2*sigma^4+36*_z1^2*beta^3*k*p^2-48*_z1^2*beta^2*p^2*sigma^2+12*_z1*beta^2*k^3*p*sigma-16*_z1*beta^2*k*q*sigma^3+8*beta^2*p^2*sigma^4+9*_z1^2*beta^3*p^2+36*_z1*beta^2*k^2*p*sigma-16*_z1*beta^2*q*sigma^3-16*_z1*beta*k*p*sigma^3+36*_z1*beta^2*k*p*sigma-16*_z1*beta*p*sigma^3+12*_z1*beta^2*p*sigma+4*beta*k^2*sigma^2+8*beta*k*sigma^2+16*sigma^4+4*beta*sigma^2)*(4*_z1^3*beta*p^3*sigma^2-6*_z1^2*beta*k*p^2*sigma+2*beta*k*p^2*sigma^3-6*_z1^2*beta*p^2*sigma+2*beta*p^2*sigma^3+3*_z1*beta*k^2*p-4*_z1*beta*q*sigma^2+6*_z1*beta*k*p-4*_z1*p*sigma^2+3*_z1*beta*p+2*k*sigma+2*sigma)^2/sigma^8, _z1 = sigma .. _z1), _z1 = sigma .. -sigma)+1)*y/sigma-(1/2)*(Int(Int(-(1/256)*G*(16*_z1^6*beta^3*p^6*sigma^4-48*_z1^5*beta^3*k*p^5*sigma^3+16*_z1^3*beta^3*k*p^5*sigma^5-48*_z1^5*beta^3*p^5*sigma^3+16*_z1^3*beta^3*p^5*sigma^5+60*_z1^4*beta^3*k^2*p^4*sigma^2-32*_z1^4*beta^3*p^3*q*sigma^4-24*_z1^2*beta^3*k^2*p^4*sigma^4+4*beta^3*k^2*p^4*sigma^6+120*_z1^4*beta^3*k*p^4*sigma^2-32*_z1^4*beta^2*p^4*sigma^4-48*_z1^2*beta^3*k*p^4*sigma^4+8*beta^3*k*p^4*sigma^6+60*_z1^4*beta^3*p^4*sigma^2-36*_z1^3*beta^3*k^3*p^3*sigma+48*_z1^3*beta^3*k*p^2*q*sigma^3-24*_z1^2*beta^3*p^4*sigma^4+12*_z1*beta^3*k^3*p^3*sigma^3-16*_z1*beta^3*k*p^2*q*sigma^5+4*beta^3*p^4*sigma^6-108*_z1^3*beta^3*k^2*p^3*sigma+48*_z1^3*beta^3*p^2*q*sigma^3+64*_z1^3*beta^2*k*p^3*sigma^3+36*_z1*beta^3*k^2*p^3*sigma^3-16*_z1*beta^3*p^2*q*sigma^5-16*_z1*beta^2*k*p^3*sigma^5-108*_z1^3*beta^3*k*p^3*sigma+64*_z1^3*beta^2*p^3*sigma^3+9*_z1^2*beta^3*k^4*p^2-24*_z1^2*beta^3*k^2*p*q*sigma^2+16*_z1^2*beta^3*q^2*sigma^4+36*_z1*beta^3*k*p^3*sigma^3-16*_z1*beta^2*p^3*sigma^5-36*_z1^3*beta^3*p^3*sigma+36*_z1^2*beta^3*k^3*p^2-48*_z1^2*beta^3*k*p*q*sigma^2-48*_z1^2*beta^2*k^2*p^2*sigma^2+32*_z1^2*beta^2*p*q*sigma^4+12*_z1*beta^3*p^3*sigma^3+8*beta^2*k^2*p^2*sigma^4+54*_z1^2*beta^3*k^2*p^2-24*_z1^2*beta^3*p*q*sigma^2-96*_z1^2*beta^2*k*p^2*sigma^2+16*_z1^2*beta*p^2*sigma^4+16*beta^2*k*p^2*sigma^4+36*_z1^2*beta^3*k*p^2-48*_z1^2*beta^2*p^2*sigma^2+12*_z1*beta^2*k^3*p*sigma-16*_z1*beta^2*k*q*sigma^3+8*beta^2*p^2*sigma^4+9*_z1^2*beta^3*p^2+36*_z1*beta^2*k^2*p*sigma-16*_z1*beta^2*q*sigma^3-16*_z1*beta*k*p*sigma^3+36*_z1*beta^2*k*p*sigma-16*_z1*beta*p*sigma^3+12*_z1*beta^2*p*sigma+4*beta*k^2*sigma^2+8*beta*k*sigma^2+16*sigma^4+4*beta*sigma^2)*(4*_z1^3*beta*p^3*sigma^2-6*_z1^2*beta*k*p^2*sigma+2*beta*k*p^2*sigma^3-6*_z1^2*beta*p^2*sigma+2*beta*p^2*sigma^3+3*_z1*beta*k^2*p-4*_z1*beta*q*sigma^2+6*_z1*beta*k*p-4*_z1*p*sigma^2+3*_z1*beta*p+2*k*sigma+2*sigma)^2/sigma^8, _z1 = sigma .. _z1), _z1 = sigma .. -sigma))+1/2

(5)

NULL


 

Download help_ode.mw

Thank you for your time. I tried in the following two way can you please have a look on it make correct@acer 

help_ode.mw

help_ode_1.mw

it is working now thanks for help@acer 

a picture of the above problem

a picture of the above problem

 

thanks alot sir @tomleslie 

and sir i fixed digits upto 8 , the residual not approaches to zero for different value of k @tomleslie 

yes, sir, I was interested in the same thing you did, but the problem is that when I fixed digits up to 6 rather than 30 the residual become flot(-Infinity). Sir, I want 4 digits after the decimal points

@tomleslie 

and sir i want to find the different value of lambda for different K (0.1,02,0.3 etc)

 

@acer 

Thank you sir, still i am confused about the guess value of lambda? how could you select these intervals for lambda?

@acer 

@Preben Alsholm Thank you for your reply. I did in the same but I want to solve by using shooting method...if the given boundary condition is not got then used P(0)=P(1)=0

Thank you sir.  The problem is that I do not know the exact policy of this plate form and I admit this is the little repeated question but it contains a new problem that I am facing. i do not receive any message can you please share it again with me@acer 

Can you please upload the maple file for above mentioned method and problem@Carl Love 

@Ronan  I am interested to find expression between Q and all other involved value after putting T=0. can you please upload maple file

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