How to solve a Bessel differential equation with a boundary condition at infinity? - Mathematica Stack Exchange most recent 30 from mathematica.stackexchange.com 2019-09-19T20:13:40Z https://mathematica.stackexchange.com/feeds/question/189158 https://creativecommons.org/licenses/by-sa/4.0/rdf https://mathematica.stackexchange.com/q/189158 2 How to solve a Bessel differential equation with a boundary condition at infinity? Adrián Silva Caballero https://mathematica.stackexchange.com/users/62030 2019-01-09T18:54:32Z 2019-04-03T23:44:01Z <p>I'm trying to solve a differential equation which solution is in the form of Bessel functions. One of the boundary conditions is at infinity. I use:</p> <pre><code>ψ[R_] = ψ[R] /. DSolve[{ψ''[R] + (ψ'[R])/R - κ^2 ψ[R] == 0, ψ[a/2] == Psi0, ψ[M] == 0}, ψ[R], R][] Limit[ψ[R], M -&gt; Infinity] Simplify[%, {r &gt; 0 &amp;&amp; κ &gt; 0 &amp;&amp; a &gt; 0 &amp;&amp; Psi0 ∈ Reals}] </code></pre> <p>And obtain:</p> <p><a href="https://i.stack.imgur.com/YLbrD.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YLbrD.png" alt="Problem"></a></p> <p>How can I obtain the solution of my problem overpassing the time limit?</p> https://mathematica.stackexchange.com/questions/189158/-/189162#189162 2 Answer by mv1996 for How to solve a Bessel differential equation with a boundary condition at infinity? mv1996 https://mathematica.stackexchange.com/users/62272 2019-01-09T19:13:32Z 2019-01-09T19:13:32Z <p>TimeConstraint helps to increase the time limit spent on Simplify operation. In case this is just a computation time issue and not an algorithmic one, this would help.</p> <p><a href="https://reference.wolfram.com/language/ref/TimeConstraint.html" rel="nofollow noreferrer">https://reference.wolfram.com/language/ref/TimeConstraint.html</a></p> https://mathematica.stackexchange.com/questions/189158/-/189176#189176 6 Answer by Bill Watts for How to solve a Bessel differential equation with a boundary condition at infinity? Bill Watts https://mathematica.stackexchange.com/users/53121 2019-01-09T22:28:44Z 2019-01-09T22:28:44Z <p>Change the order of execution.</p> <pre><code>ψ[R_] = ψ[R] /. DSolve[{ψ''[R] + (Derivative[ψ][R])/ R - κ^2 ψ[R] == 0, ψ[a/2] == Psi0, ψ[M] == 0}, ψ[R], R][] \$Assumptions = r &gt; 0 &amp;&amp; κ &gt; 0 &amp;&amp; a &gt; 0 &amp;&amp; Psi0 ∈ Reals &amp;&amp; M &gt; 0 &amp;&amp; R &gt; 0 ψ[R_] = FullSimplify[ψ[R]] (*(Psi0 (BesselI[0, M κ] BesselK[0, R κ] - BesselK[0, M κ] BesselI[0, R κ]))/( BesselK[0, (a κ)/2] BesselI[0, M κ] - BesselI[0, (a κ)/2] BesselK[0, M κ])*) Limit[ψ[R], M -&gt; ∞] (*(Psi0 BesselK[0, R κ])/BesselK[0, (a κ)/2]*) </code></pre>