mehdi jafari

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These are questions asked by mehdi jafari

Hi. i have a nonlinear integral equation. any idea for solving it? tnx in advance

integral-equation.mw

 

i have a nonlinear equation,W, which have two unknowns: x and b 
i want to solve W for b, and the x domain is : -4e-6<x<4e-6. can anyone help? tnx in advance


 

restart; with(DirectSearch)

E := (-7.29511346879067*10^9*b-12693.49752)*((D@@2)(phi))(x)+(5.36829162574305*b+0.3487242314e-3)*((D@@4)(phi))(x)+(2.02316788462733*10^17*b+4.107030812*10^12)*phi(x)+1.74615339888993*10^17*b

(-7295113469.*b-12693.49752)*((D@@2)(phi))(x)+(5.36829162574305*b+0.3487242314e-3)*((D@@4)(phi))(x)+(0.2023167885e18*b+0.4107030812e13)*phi(x)+0.1746153399e18*b

(1)

dsolve(E, phi(x))

ivp := {E, phi(-0.406e-3) = 1, phi(0.406e-3) = 1, (D(phi))(-0.406e-3) = 0, (D(phi))(0.406e-3) = 0}

{(-7295113469.*b-12693.49752)*((D@@2)(phi))(x)+(5.36829162574305*b+0.3487242314e-3)*((D@@4)(phi))(x)+(0.2023167885e18*b+0.4107030812e13)*phi(x)+0.1746153399e18*b, phi(-0.406e-3) = 1, phi(0.406e-3) = 1, (D(phi))(-0.406e-3) = 0, (D(phi))(0.406e-3) = 0}

(2)

N := dsolve(ivp, phi(x))

A := simplify(rhs(dsolve(ivp, phi(x))))

W := evalf(subs(phi(x) = simplify(rhs(dsolve(ivp, phi(x)))), (2.74505742817409*10^8+3.18054074382591*10^8*phi(x))^2-(1.63833629390402*(2.74505742817409*10^8+3.18054074382591*10^8*phi(x)))*(diff(phi(x), x, x))+3*(6.98727355657655*10^7+32282.4885498329*(diff(phi(x), x)))^2+2.68414581287153*(diff(phi(x), x, x))^2 = 0.9e17))

 

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Download MZ2.mw

how this integro-differential equation can be solved?
any assumption or suggestion is appreciated. tnx in advance

EDITED: parameters are added.


 

restart

w[c]:=0.01:delta:=5:Omega:=5:w[0]:=1:gamma_0:=1:k[b]:=1:T:=100:nu[n]:=2*Pi*n*k[b]*T;

200*Pi*n

(1)

F:=2*gamma_0*w[c]^2*exp(-w[c]*tau)*sin(delta/Omega*(sin(Omega*t)-sin(Omega*(t-tau)))+w[0]*tau):

k1:=2*k[b]*T*w[c]^2*sum((w[c]*exp(-w[c]*tau)-abs(nu[n])*exp(-abs(nu[n])*tau))/(w[c]^2-nu[n]^2),n=-infinity..infinity):

G:=cos(delta/Omega*(sin(Omega*t)-sin(Omega*(t-tau)))+w[0]*tau)*k1:

eq:=diff(rho(t),t)+Int(G*rho(tau),tau=0..t)=1/2*Int(G-F,tau=0..t);

diff(rho(t), t)+Int(0.200e-1*cos(sin(5*t)-sin(5*t-5*tau)+tau)*(sum((0.1000000000e-1*exp(-0.1000000000e-1*tau)-628.3185308*abs(n)*exp(-628.3185308*abs(n)*tau))/(-394784.1762*n^2+0.1000000000e-3), n = -infinity .. infinity))*rho(tau), tau = 0 .. t) = (1/2)*(Int(0.200e-1*cos(sin(5*t)-sin(5*t-5*tau)+tau)*(sum((0.1000000000e-1*exp(-0.1000000000e-1*tau)-628.3185308*abs(n)*exp(-628.3185308*abs(n)*tau))/(-394784.1762*n^2+0.1000000000e-3), n = -infinity .. infinity))-0.2e-3*exp(-0.1e-1*tau)*sin(sin(5*t)-sin(5*t-5*tau)+tau), tau = 0 .. t))

(2)

 


 

Download integro.mw

i have ode system of two dependent variables y and x  and one independent varibale time. how can i find explicit expression between y and x? 

 

restart

sys:=diff(x(t),t)=x(t)*y(t)+t,diff(y(t),t)=x(t)-t;

diff(x(t), t) = x(t)*y(t)+t, diff(y(t), t) = x(t)-t

(1)

dsolve([sys,x(0)=0,y(0)=1],[x(t),y(t)],series)

{x(t) = series((1/2)*t^2+(1/6)*t^3+(1/24)*t^4-(1/24)*t^5+O(t^6),t,6), y(t) = series(1-(1/2)*t^2+(1/6)*t^3+(1/24)*t^4+(1/120)*t^5+O(t^6),t,6)}

(2)

 

 

 


 

Download explicit.mw

I have a system of six ODEs. i solve them numerically using dolve,numeric command. the problem is setting step size for example stepsize=1e-5 or minstep=500 lead to a result of order 1e-2; But without using this option, results are of order integer numbers. could any one help? tnx in advance

EDITED
 

Download ode_problem.mw

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