salim-barzani

Seeking Insights into a Novel Approach f...

Asked: salim-barzani 1785 Product: Maple 2024

December 30 2024

1 5

I’ve spent considerable effort trying to understand how the solution was derived, particularly the approach involving the factoring of G′/G. Despite my attempts, the methodology remains elusive. It seems there’s an innovative idea at play here—something beyond the techniques we’ve applied in similar problems before. While I suspect it involves a novel perspective, I can’t quite pinpoint what it might be.

If anyone has insights into how this factoring is achieved or can shed light on the underlying idea, I’d greatly appreciate your help.

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Warning, A new binding for the name `gamma` has been created. The global instance of this name is still accessible using the :- prefix, :-`gamma`. See ?protect for details.

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F := sum(c[i]*(m+(diff(G(xi), xi))/G(xi))^i, i = -1 .. 1)

c[-1]/(m+(diff(G(xi), xi))/G(xi))+c[0]+c[1]*(m+(diff(G(xi), xi))/G(xi))

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-c[-1]*((diff(diff(G(xi), xi), xi))/G(xi)-(diff(G(xi), xi))^2/G(xi)^2)/(m+(diff(G(xi), xi))/G(xi))^2+c[1]*((diff(diff(G(xi), xi), xi))/G(xi)-(diff(G(xi), xi))^2/G(xi)^2)

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-c[-1]*((-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/G(xi)-(diff(G(xi), xi))^2/G(xi)^2)/(m+(diff(G(xi), xi))/G(xi))^2+c[1]*((-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/G(xi)-(diff(G(xi), xi))^2/G(xi)^2)

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NULL

>	# rewritting rule RR := isolate(m+diff(G(xi), xi)/(G(xi))=Phi, diff(G(xi), xi)/G(xi));

(diff(G(xi), xi))/G(xi) = Phi-m

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>	# Apply RR and collect wrt Phi subs(RR, L): normal(%): PhiN := collect(numer(lhs(%)), phi): PhiD := denom(lhs(%%));

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>	with(LargeExpressions): LLE := collect(PhiN, Phi, Veil[phi] ): LLE / PhiD = 0;

(Phi^6*phi[1]+3*Phi^5*phi[2]-Phi^4*phi[3]-Phi^3*phi[4]-Phi^2*phi[5]+Phi*phi[6]-phi[7])/(Phi^3*G(xi)^4) = 0

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>	# phi[i] coefficients phis := [ seq( phi[i] = simplify(Unveil[phi](phi[i]), size), i=1..LastUsed[phi] ) ]: print~( phis ):

phi[3] = -3*G(xi)^4*c[1]*(-(1/3)*a[1]*k^2+(-c[-1]*(gamma-sigma)*c[1]+(-gamma+sigma)*c[0]^2+(1/3)*w*a[2]-(1/3)*alpha)*k+b*c[-1]*c[1]+b*c[0]^2-(1/3)*w)

phi[4] = G(xi)*(2*c[1]*(V*a[2]-a[1])*(diff(G(xi), xi))^3+3*c[1]*G(xi)*(2*m*mu+lambda)*(V*a[2]-a[1])*(diff(G(xi), xi))^2+((2*m*mu+lambda)^2*G(xi)+3*mu)*(V*a[2]-a[1])*G(xi)*c[1]*(diff(G(xi), xi))+G(xi)^2*(-c[0]*(6*c[-1]*((-gamma+sigma)*k+b)*c[1]-a[1]*k^2+k*w*a[2]+((-gamma+sigma)*k+b)*c[0]^2-k*alpha-w)*G(xi)+c[1]*mu*(2*m*mu+lambda)*(V*a[2]-a[1])))

phi[5] = -3*G(xi)^4*(-(1/3)*a[1]*k^2+(-c[-1]*(gamma-sigma)*c[1]+(-gamma+sigma)*c[0]^2+(1/3)*w*a[2]-(1/3)*alpha)*k+b*c[-1]*c[1]+b*c[0]^2-(1/3)*w)*c[-1]

phi[6] = 4*c[-1]*((1/2)*(V*a[2]-a[1])*(diff(G(xi), xi))^3+(3/2)*(m*mu+(1/2)*lambda)*(V*a[2]-a[1])*G(xi)*(diff(G(xi), xi))^2+(V*a[2]-a[1])*((m*mu+(1/2)*lambda)^2*G(xi)+(3/4)*mu)*G(xi)*(diff(G(xi), xi))+(1/2)*G(xi)^2*((3/2)*c[-1]*((-gamma+sigma)*k+b)*c[0]*G(xi)+(m*mu+(1/2)*lambda)*(V*a[2]-a[1])*mu))*G(xi)

phi[7] = 8*((1/4)*(V*a[2]-a[1])*(diff(G(xi), xi))^4+(V*a[2]-a[1])*G(xi)*(m*mu+(1/2)*lambda)*(diff(G(xi), xi))^3+(V*a[2]-a[1])*G(xi)*((m*mu+(1/2)*lambda)^2*G(xi)+(1/2)*mu)*(diff(G(xi), xi))^2+(V*a[2]-a[1])*G(xi)^2*(m*mu+(1/2)*lambda)*mu*(diff(G(xi), xi))+(1/4)*G(xi)^2*(-(1/2)*((-gamma+sigma)*k+b)*c[-1]^2*G(xi)^2+mu^2*(V*a[2]-a[1])))*c[-1]

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>	# WATCHOUT: you have 9 coefficients and so its desirable to have the same number of unknowns unknowns := indets(rhs~(phis), name); COEFFS := solve(rhs~(phis), unknowns)

Error, (in solve) cannot solve expressions with diff(G(xi),xi) for xi

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>	case1 := COEFFS[4]

${alpha = alpha, beta = gamma, delta = delta, gamma = gamma, k = k, lambda = 0, m = 2*n, mu = mu, n = n, sigma = 32*alpha*mu^2*n^4/a[-1]^2, w = -2*alpha*k^2*n-4*alpha*mu^2*n+delta^2, a[-1] = a[-1], a[0] = 0, a[1] = 0}$

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a[-1]/(2*n+1/(diff(G(xi), xi)))

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128*V(xi)^4*n^6*alpha*mu^2/a[-1]^2+(16*alpha*k^2*n^4-8*delta^2*n^3+8*n^3*(-2*alpha*k^2*n-4*alpha*mu^2*n+delta^2))*V(xi)^2-4*V(xi)*(diff(diff(V(xi), xi), xi))*alpha*n^2 = 0

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V(xi) = a[-1]/(2*n+1/(diff(G(xi), xi)))

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E := diff(G(xi), xi) = -(-2*m*mu-lambda)*exp(-(2*m*mu+lambda)*xi)*c__1/(2*m*mu+lambda)-mu/(2*m*mu+lambda)

diff(G(xi), xi) = -(-2*m*mu-lambda)*exp(-(2*m*mu+lambda)*xi)*c__1/(2*m*mu+lambda)-mu/(2*m*mu+lambda)

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V(xi) = a[-1]/(2*n+1/(-(-2*m*mu-lambda)*exp(-(2*m*mu+lambda)*xi)*c__1/(2*m*mu+lambda)-mu/(2*m*mu+lambda)))

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V(xi) = a[-1]/(2*n+1/(exp(-4*mu*n*xi)*c__1-(1/4)/n))

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a[-1]/(2*n+1/(exp(-4*mu*n*xi)*c__1-(1/4)/n))

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a[-1]/(2*n+1/((cosh(4*mu*n*xi)-sinh(4*mu*n*xi))*c__1-(1/4)/n))

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a[-1]/(2*n+1/((cosh(4*mu*n*xi)-sinh(4*mu*n*xi))*c__1-(1/4)/n))

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Download problem99.mw

Issue with Solving Individual Parameters...

Asked: salim-barzani 1785 Product: Maple

December 30 2024

1 2

When I attempt to solve for a specific parameter in Maple, it results in an error stating that the parameter does not exist. However, if I solve for all parameters, it successfully finds them, as shown in the figure. How can I resolve this issue?

parameter.mw

why taking double derivative with two di...

Asked: salim-barzani 1785 Product: Maple 2024

December 27 2024

0 6

I did a lot of time but this time i don't know why not run any one have idea?

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Download why.mw

How factoring two term after substituti...

Asked: salim-barzani 1785 Product: Maple 2024

December 27 2024

1 6

I already get the same results, but there's something about the factoring process that I encountered for the first time in this ODE. In the paper, it says that G′ satisfies a certain condition, but I’m not sure exactly what that means. Did the author use it for substitution, or did they change (m+F) into another variable and then solve? I’m not exactly sure what approach was taken. Does anyone have any idea or insight into this?

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{t = tau, x = (-Zeta*c[3]-tau*c[4]-`Υ`*c[2]+xi)/c[1], y = `Υ`, z = Zeta, u(x, y, z, t) = U(xi)}

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F := sum(a[i]*(m+1/(diff(G(xi), xi)))^i, i = -1 .. 1)

a[-1]/(m+1/(diff(G(xi), xi)))+a[0]+a[1]*(m+1/(diff(G(xi), xi)))

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a[-1]*(diff(diff(G(xi), xi), xi))/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(diff(diff(G(xi), xi), xi))/(diff(G(xi), xi))^2

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a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2

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U(xi) = a[-1]/(m+1/(diff(G(xi), xi)))+a[0]+a[1]*(m+1/(diff(G(xi), xi)))

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diff(U(xi), xi) = a[-1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/((m+1/(diff(G(xi), xi)))^2*(diff(G(xi), xi))^2)-a[1]*(-(2*m*mu+lambda)*(diff(G(xi), xi))-mu)/(diff(G(xi), xi))^2