# Solving infinite coupled equations

I need to solve symbolically an infinite system of coupled algebraic equations that I tried to do analytically but I could not. Solutions of these equations, $$V_l^m$$, define the coefficients of a some function $$V$$ (that I'm looking for) of the form

$$V(r,\theta ,\varphi)=\sum _{l=0}^{\infty } \sum _{m=-l}^{+l} Y_l^m (\theta ,\varphi) V_l^m(r)=Y_0^0 V_0^0+Y_1^0 V_1^0+Y_1^1 V_1^1+Y_1^{-1} V_1^{-1}+\cdots$$

where $$Y_l^m (\theta ,\varphi)$$ denote the spherical harmonics.

These solutions $$V_l^m$$ obey the general form

$$A\, V_l^m+\frac{B}{2}\left(-\sqrt{\frac{(l+m-1) (l+m)}{(2 l-1) (2 l+1)}} V_{l-1}^{m-1}+\sqrt{\frac{(l-m) (l-m-1)}{(2 l+1) (2 l-1)}} V_{l-1}^{m+1}+\sqrt{\frac{(l-m+1) (l-m+2)}{(2 l+1) (2 l+3)}} V_{l+1}^{m-1}-\sqrt{\frac{(l+m+2) (l+m+1)}{(2 l+1) (2 l+3)}} V_{l+1}^{m+1}\right)+m R V_l^m+\left(\delta _{l,1} \delta _{m,-1}-\delta _{l,1} \delta _{m,1}\right)D\, \lambda =0$$

which involves an infinite coupled equations.

Here $$A, B, D, R, \lambda$$ are real constants.

In MA input form it gives

eq[l_,m_]:=A Subscript[V,{l,m}]+B/2 (Sqrt[((l-m+1)(l-m+2))/((2l+1)(2l+3))] Subscript[V,{l+1,m-1}]+Sqrt[((l-m)(l-m-1))/((2l+1)(2l-1))] Subscript[V,{l-1,m+1}]-Sqrt[((l+m-1)(l+m))/((2l-1)(2l+1))] Subscript[V,{l-1,m-1}]-Sqrt[((l+m+2)(l+m+1))/((2l+1)(2l+3))] Subscript[V,{l+1,m+1}])+(KroneckerDelta[l,1]KroneckerDelta[m,-1]-KroneckerDelta[l,1]KroneckerDelta[m,1])D \[Lambda]+m R Subscript[V,{l,m}]


Here just for the code I used $$V_{\{l,m\}}$$ instead of $$V_l^m$$.

For your information, solution of this problem is to express $$V_{\{0,0\}}$$ according to the variable $$\lambda$$ as

$$V_{\{0,0\}}=(f)\lambda$$

where $$f$$ is a term that includes all other solutions (maybe in the form of an infinite continued fraction).

Any help please on this problem! Can MA do something?

• With eqs defined as the list of all equations explicitly given above, the number of distinct coefficients V is Length@Union@Cases[eqs, {_, _}, Infinity], which evaluates to 18, while the number of equations is Length@eqs, which evaluates to 10. Hence, the coefficients V in these equations are underdetermined. Of course, V in a larger set of equations might not be underdetermined, but it is impossible to say without knowing the general form of the equations. By the way use == instead of = in the equations to avoid syntax errors. – bbgodfrey Aug 19 at 17:28
• @bbgodfrey, that is the problem for a coupled equations, so is there a possibility of having the solution if the general form is known? – Gallagher Aug 19 at 18:27
• I just edited the question. – Gallagher Aug 19 at 19:40

# first attempt

(apparently this isn't what the OP wanted)

I have edited the equation a bit because D is a reserved symbol and because subscripts are inconvenient:

eq[l_,m_] = d λ (KroneckerDelta[-1, m] KroneckerDelta[1, l] - KroneckerDelta[1, l] KroneckerDelta[1, m]) + a V[l, m] +
m R V[l, m] +
1/2 b (-Sqrt[(((-1 + l + m) (l + m))/((-1 + 2 l) (1 + 2 l)))] V[l-1, m-1] +
Sqrt[((-1 + l - m) (l - m))/((-1 + 2 l) (1 + 2 l))] V[l-1, m+1] +
Sqrt[((1 + l - m) (2 + l - m))/((1 + 2 l) (3 + 2 l))] V[l+1, m-1] -
Sqrt[((1 + l + m) (2 + l + m))/((1 + 2 l) (3 + 2 l))] V[l+1, m+1]);


For every value of l, we have 2l+1 equations that can be used to determine {V[l+1,-l-1],...,V[l+1,l+1]}. There are 2l+3 unknowns, so we cannot determine all coefficients. Here I've chosen to solve for the {V[l+1,-l-1],...,V[l+1,l-1]} and leave V[l+1,l] and V[l+1,l+1] as free parameters:

S[l_] := First@Solve[Table[eq[l, m] == 0, {m, -l, l}],
Table[V[l+1, m], {m, -l-1, l-1}]]


Try it out:

S
(*    {V[3, -3] -> ...,
V[3, -2] -> ...,
V[3, -1] -> ...,
V[3, 0] -> ...,
V[3, 1] -> ...}    *)


We can stack these solutions in order to express the coefficients at a given value of l in terms of all smaller l-values:

Expand[S /. S /. S]
(*    complicated expression for V[3,-3]...V[3,1] in terms of
V[0, 0], V[1, 0], V[1, 1], V[2, 1], V[2, 2], V[3, 2], V[3, 3]}    *)


programmatically:

Clear[F];
F[l_] := Module[{f = S[l]},
Do[f = f /. S[L], {L, l - 1, 0, -1}];
Expand[f]]


Now calling F gives the same result as the above Expand[S /. S /. S].

# second attempt

If you want to express $$V_{0,0}$$ in terms of $$\{V_{L,-L},V_{L,-L+1},\ldots,V_{L,L}\}$$ for a specific $$L$$, you can solve all the equations for $$l for all the variables with $$l. For example, with $$L=4$$ we express $$V_{0,0}$$ in terms of $$\{V_{4,-4},V_{4,-3},V_{4,-2},V_{4,-1},V_{4,0},V_{4,1},V_{4,2},V_{4,3},V_{4,4}\}$$:

With[{L = 4},
FullSimplify[V[0, 0] /.
First@Solve[
Flatten[Table[eq[l, m] == 0, {l, 0, L - 1}, {m, -l, l}]],
Flatten[Table[V[l, m],       {l, 0, L - 1}, {m, -l, l}]]]]]


-((b (-10290 Sqrt a^7 d λ + 42 Sqrt a^2 b^3 R (35 Sqrt R^2 (-V[4, -2] + V[4, 2]) + 2 b^2 (7 V[4, -4] - Sqrt V[4, -2] + Sqrt V[4, 2] - 7 V[4, 4])) - 294 Sqrt a^4 b^3 R (7 V[4, -4] - Sqrt V[4, -2] + Sqrt V[4, 2] - 7 V[4, 4]) + 6 Sqrt b^3 R (21 Sqrt b^2 R^2 (V[4, -2] - V[4, 2]) + 49 R^4 (7 V[4, -4] + 4 Sqrt V[4, -2] - 4 Sqrt V[4, 2] - 7 V[4, 4]) + b^4 (-7 V[4, -4] + Sqrt V[4, -2] - Sqrt V[4, 2] + 7 V[4, 4])) + 49 a^5 (174 Sqrt b^2 d λ + 2940 Sqrt d R^2 λ + b^3 (-7 Sqrt V[4, -4] + 2 Sqrt V[4, -2] - 3 Sqrt V[4, 0] + 2 Sqrt V[4, 2] - 7 Sqrt V[4, 4])) - 14 a^3 (153 Sqrt b^4 d λ + 1680 Sqrt b^2 d R^2 λ + 36015 Sqrt d R^4 λ + 35 Sqrt b^3 R^2 (7 Sqrt V[4, -4] + Sqrt V[4, -2] - 3 Sqrt V[4, 0] + Sqrt V[4, 2] + 7 Sqrt V[4, 4]) - 2 b^5 (7 Sqrt V[4, -4] - 2 Sqrt V[4, -2] + 3 Sqrt V[4, 0] - 2 Sqrt V[4, 2] + 7 Sqrt V[4, 4])) + a (162 Sqrt b^6 d λ + 252 Sqrt b^4 d R^2 λ + 50274 Sqrt b^2 d R^4 λ + 370440 Sqrt d R^6 λ + 14 Sqrt b^5 R^2 (28 Sqrt V[4, -4] + Sqrt V[4, -2] - 6 Sqrt V[4, 0] + Sqrt V[4, 2] + 28 Sqrt V[4, 4]) + 49 Sqrt b^3 R^4 (77 Sqrt V[4, -4] + 8 Sqrt V[4, -2] - 27 Sqrt V[4, 0] + 8 Sqrt V[4, 2] + 77 Sqrt V[4, 4]) - 3 b^7 (7 Sqrt V[4, -4] - 2 Sqrt V[4, -2] + 3 Sqrt V[4, 0] - 2 Sqrt V[4, 2] + 7 Sqrt V[4, 4]))))/(3 Sqrt[ 42] (1715 a^9 - 1225 a^7 (2 b^2 + 21 R^2) + 7 a^5 (156 b^4 + 2380 b^2 R^2 + 15435 R^4) - a^3 (174 b^6 + 3045 b^4 R^2 + 31850 b^2 R^4 + 145775 R^6) + 3 a (3 b^8 + 58 b^6 R^2 + 651 b^4 R^4 + 5880 b^2 R^6 + 20580 R^8))))

You can do this for larger values of $$L$$, but it gets more difficult.

• Roman, thank's for this approach, but the idea is not to calculate all solutions, it's impossible, so I try to express only V[0,0] according to the others solutions. I know that MA can do it. – Gallagher Aug 19 at 21:44
• If you know that Mathematica can do it, then why don't you tell us? Why are you asking? – Roman Aug 19 at 21:50
• Sorry Roman, I'm new with Mathematica, I think that Mathematica is very powerful. – Gallagher Aug 19 at 21:54
• I think that it's possible to proceed by truncation at different coefficient by dropping the higher order coefficients in order to build a symbolic solution of the form V[0,0] = f(V[l+1,m+1],...) = f(f(V[l,m],...)) = f(f(f(V[l-1,m-1],...))) = f(f(.......f(V[1,1]))) \lambda, so how do it with Mathematica? – Gallagher Aug 20 at 14:01
• Apparently it's impossible to compute the function $V(r,\theta,\varphi)$? – Gallagher Aug 20 at 16:56