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Question: How to solve coupled nonlinear differential equations numerically?

Question:How to solve coupled nonlinear differential equations numerically?

hendriksdf5 50 Maple
ode dsolve numeric differential-equations
July 30 2024
1

I would like to solve this system of differential equations y_1 , y_2. However, there should be no exact solutions for this problem. Is there a way to get a numerical solution of such coupled equations in maple and if so, how? I know the dsolve() command, but it did not work here. The boundary conditions are f(0)-1 = K(0) = 0 and f'(inf) = K'(inf) = 0. It would be nice if someone could help me or tell me where to look to solve something like this.

restart

with(Physics)NULL

[coordinatesystems = {X}, signature = `+ + + -`, spaceindices = lowercaselatin, spacetimeindices = greek, su2indices = uppercaselatin]

 (1)

``Setup(realobjects = {a, g, v, K(diff(rho(x), x)), f(diff(rho(x), x))})

[realobjects = {a, g, v, K(`rho'`), f(`rho'`)}]

 (2)

-(`rho'`^2*(diff(K(`rho'`), `rho'`, `rho'`))+2*`rho'`*(diff(K(`rho'`), `rho'`))+K(`rho'`)*(K(`rho'`)^2*a*`rho'`^2*v^2+(1/2)*f(`rho'`)^2+(-2*`rho'`-1)*f(`rho'`)-a*v^2*`rho'`^2+2*`rho'`-3/2))*v^2 = 0

-(`rho'`^2*(diff(diff(K(`rho'`), `rho'`), `rho'`))+2*(diff(K(`rho'`), `rho'`))*`rho'`+K(`rho'`)*(K(`rho'`)^2*a*`rho'`^2*v^2+(1/2)*f(`rho'`)^2+(-2*`rho'`-1)*f(`rho'`)-a*v^2*`rho'`^2+2*`rho'`-3/2))*v^2 = 0

 (3)

2*(diff(f(`rho'`), `rho'`, `rho'`))/g^2+(-4*f(`rho'`)^3+24*f(`rho'`)^2+(-`rho'`^2*v^2*g^2*K(`rho'`)^2-44)*f(`rho'`)+24+(2*(`rho'`+1/2))*g^2*`rho'`^2*v^2*K(`rho'`)^2)/(2*`rho'`^2*g^2) = 0

2*(diff(diff(f(`rho'`), `rho'`), `rho'`))/g^2+(1/2)*(-4*f(`rho'`)^3+24*f(`rho'`)^2+(-`rho'`^2*v^2*g^2*K(`rho'`)^2-44)*f(`rho'`)+24+2*(`rho'`+1/2)*g^2*`rho'`^2*v^2*K(`rho'`)^2)/(`rho'`^2*g^2) = 0

 (4)

y_1 := -(`rho'`^2*(diff(diff(K(`rho'`), `rho'`), `rho'`))+2*`rho'`*(diff(K(`rho'`), `rho'`))+K(`rho'`)*(K(`rho'`)^2*a*`rho'`^2*v^2+(1/2)*f(`rho'`)^2+(-2*`rho'`-1)*f(`rho'`)-a*v^2*`rho'`^2+2*`rho'`-3/2))*v^2 = 0

-(`rho'`^2*(diff(diff(K(`rho'`), `rho'`), `rho'`))+2*(diff(K(`rho'`), `rho'`))*`rho'`+K(`rho'`)*(K(`rho'`)^2*a*`rho'`^2*v^2+(1/2)*f(`rho'`)^2+(-2*`rho'`-1)*f(`rho'`)-a*v^2*`rho'`^2+2*`rho'`-3/2))*v^2 = 0

 (5)

y_2 := 2*(diff(diff(f(`rho'`), `rho'`), `rho'`))/g^2+(1/2)*(-4*f(`rho'`)^3+24*f(`rho'`)^2+(-`rho'`^2*v^2*g^2*K(`rho'`)^2-44)*f(`rho'`)+24+2*(`rho'`+1/2)*g^2*`rho'`^2*v^2*K(`rho'`)^2)/(`rho'`^2*g^2) = 0

2*(diff(diff(f(`rho'`), `rho'`), `rho'`))/g^2+(1/2)*(-4*f(`rho'`)^3+24*f(`rho'`)^2+(-`rho'`^2*v^2*g^2*K(`rho'`)^2-44)*f(`rho'`)+24+2*(`rho'`+1/2)*g^2*`rho'`^2*v^2*K(`rho'`)^2)/(`rho'`^2*g^2) = 0

 (6)

NULL

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