Solving Differential Equation With Bessel Functions












0












$begingroup$


trying to solve




$$frac{d}{dr}bigg(frac{1}{r} frac{d}{dr}(rbar u_r)bigg) - k^2 bar u_r = frac{k^2}{omega^2}phi(r) bar u_r $$




subject to




$$R_1 leq r leq R_2,$$
$$bar u_r(R_1) = bar u_r(R_2) = 0,$$




where




$$phi (r) = frac{2V}{r^2}frac{d}{dr}(rV).$$




I am trying to solve for $bar u_r $ in the case where $V(r) = 0.$



My attempt has got me to




$$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg).$$




When I try to solve this I end up with a zero solution? WolframAlpha is giving me solutions to this as Bessel functions. Could anyone point me in the right direction?



I should note, this is a fluid dynamics problem.






ordinary-differential-equations fluid-dynamics







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    0












    $begingroup$


    trying to solve




    $$frac{d}{dr}bigg(frac{1}{r} frac{d}{dr}(rbar u_r)bigg) - k^2 bar u_r = frac{k^2}{omega^2}phi(r) bar u_r $$




    subject to




    $$R_1 leq r leq R_2,$$
    $$bar u_r(R_1) = bar u_r(R_2) = 0,$$




    where




    $$phi (r) = frac{2V}{r^2}frac{d}{dr}(rV).$$




    I am trying to solve for $bar u_r $ in the case where $V(r) = 0.$



    My attempt has got me to




    $$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg).$$




    When I try to solve this I end up with a zero solution? WolframAlpha is giving me solutions to this as Bessel functions. Could anyone point me in the right direction?



    I should note, this is a fluid dynamics problem.






    ordinary-differential-equations fluid-dynamics







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      trying to solve




      $$frac{d}{dr}bigg(frac{1}{r} frac{d}{dr}(rbar u_r)bigg) - k^2 bar u_r = frac{k^2}{omega^2}phi(r) bar u_r $$




      subject to




      $$R_1 leq r leq R_2,$$
      $$bar u_r(R_1) = bar u_r(R_2) = 0,$$




      where




      $$phi (r) = frac{2V}{r^2}frac{d}{dr}(rV).$$




      I am trying to solve for $bar u_r $ in the case where $V(r) = 0.$



      My attempt has got me to




      $$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg).$$




      When I try to solve this I end up with a zero solution? WolframAlpha is giving me solutions to this as Bessel functions. Could anyone point me in the right direction?



      I should note, this is a fluid dynamics problem.






      ordinary-differential-equations fluid-dynamics







      share|cite|improve this question









      $endgroup$




      trying to solve




      $$frac{d}{dr}bigg(frac{1}{r} frac{d}{dr}(rbar u_r)bigg) - k^2 bar u_r = frac{k^2}{omega^2}phi(r) bar u_r $$




      subject to




      $$R_1 leq r leq R_2,$$
      $$bar u_r(R_1) = bar u_r(R_2) = 0,$$




      where




      $$phi (r) = frac{2V}{r^2}frac{d}{dr}(rV).$$




      I am trying to solve for $bar u_r $ in the case where $V(r) = 0.$



      My attempt has got me to




      $$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg).$$




      When I try to solve this I end up with a zero solution? WolframAlpha is giving me solutions to this as Bessel functions. Could anyone point me in the right direction?



      I should note, this is a fluid dynamics problem.






      ordinary-differential-equations fluid-dynamics




      ordinary-differential-equations fluid-dynamics






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 16 at 15:47









      rodger_kicksrodger_kicks

      9610




      9610






















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          $begingroup$

          $$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg)=0$$



          Knowing the properties of the Bessel functions and related differential equations, it is obvious that the solution is :
          $$bar u_r(r)=c_1 I_k(r)+c_2 K_k(r)$$
          The functions $I_k(r)$ and $K_k(r)$ are the Modified Bessel functions of first and second kind respectively.



          http://mathworld.wolfram.com/ModifiedBesselDifferentialEquation.html



          http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html



          http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html



          There is nothing to prove since this is the direct consequence of the definition of the Bessel functions and of the basic properties.



          Of course, $bar u_r(r)=0$ is a particular solution which satisfies the ODE (case $c_1=c_2=0$ ).



          In the above general solution $c_1$ and $c_2$ are arbitrary constants. Those constants can be determined if some well posed boundary conditions are specified.



          If in trying to solve the PDE you end up with the zero solution only, several causes can be suspected. A possible one could be that you introduces some "hided" or "implicit" conditions which are not in the wording of the problem. All depends on what method of solving you use and what exactly is your calculus. Without checking all steps of your calculus and without seeing all your intermediate equations it is impossible to give a definitive diagnosis.






          share|cite|improve this answer









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            $begingroup$

            $$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg)=0$$



            Knowing the properties of the Bessel functions and related differential equations, it is obvious that the solution is :
            $$bar u_r(r)=c_1 I_k(r)+c_2 K_k(r)$$
            The functions $I_k(r)$ and $K_k(r)$ are the Modified Bessel functions of first and second kind respectively.



            http://mathworld.wolfram.com/ModifiedBesselDifferentialEquation.html



            http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html



            http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html



            There is nothing to prove since this is the direct consequence of the definition of the Bessel functions and of the basic properties.



            Of course, $bar u_r(r)=0$ is a particular solution which satisfies the ODE (case $c_1=c_2=0$ ).



            In the above general solution $c_1$ and $c_2$ are arbitrary constants. Those constants can be determined if some well posed boundary conditions are specified.



            If in trying to solve the PDE you end up with the zero solution only, several causes can be suspected. A possible one could be that you introduces some "hided" or "implicit" conditions which are not in the wording of the problem. All depends on what method of solving you use and what exactly is your calculus. Without checking all steps of your calculus and without seeing all your intermediate equations it is impossible to give a definitive diagnosis.






            share|cite|improve this answer









            $endgroup$


















              1












              $begingroup$

              $$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg)=0$$



              Knowing the properties of the Bessel functions and related differential equations, it is obvious that the solution is :
              $$bar u_r(r)=c_1 I_k(r)+c_2 K_k(r)$$
              The functions $I_k(r)$ and $K_k(r)$ are the Modified Bessel functions of first and second kind respectively.



              http://mathworld.wolfram.com/ModifiedBesselDifferentialEquation.html



              http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html



              http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html



              There is nothing to prove since this is the direct consequence of the definition of the Bessel functions and of the basic properties.



              Of course, $bar u_r(r)=0$ is a particular solution which satisfies the ODE (case $c_1=c_2=0$ ).



              In the above general solution $c_1$ and $c_2$ are arbitrary constants. Those constants can be determined if some well posed boundary conditions are specified.



              If in trying to solve the PDE you end up with the zero solution only, several causes can be suspected. A possible one could be that you introduces some "hided" or "implicit" conditions which are not in the wording of the problem. All depends on what method of solving you use and what exactly is your calculus. Without checking all steps of your calculus and without seeing all your intermediate equations it is impossible to give a definitive diagnosis.






              share|cite|improve this answer









              $endgroup$
















                1












                1








                1





                $begingroup$

                $$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg)=0$$



                Knowing the properties of the Bessel functions and related differential equations, it is obvious that the solution is :
                $$bar u_r(r)=c_1 I_k(r)+c_2 K_k(r)$$
                The functions $I_k(r)$ and $K_k(r)$ are the Modified Bessel functions of first and second kind respectively.



                http://mathworld.wolfram.com/ModifiedBesselDifferentialEquation.html



                http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html



                http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html



                There is nothing to prove since this is the direct consequence of the definition of the Bessel functions and of the basic properties.



                Of course, $bar u_r(r)=0$ is a particular solution which satisfies the ODE (case $c_1=c_2=0$ ).



                In the above general solution $c_1$ and $c_2$ are arbitrary constants. Those constants can be determined if some well posed boundary conditions are specified.



                If in trying to solve the PDE you end up with the zero solution only, several causes can be suspected. A possible one could be that you introduces some "hided" or "implicit" conditions which are not in the wording of the problem. All depends on what method of solving you use and what exactly is your calculus. Without checking all steps of your calculus and without seeing all your intermediate equations it is impossible to give a definitive diagnosis.






                share|cite|improve this answer









                $endgroup$



                $$frac{d^2 bar u_r}{dr^2} + frac{1}{r} frac{d bar u_r}{dr} - bar u_r bigg(frac{1}{r^2}+k^2 bigg)=0$$



                Knowing the properties of the Bessel functions and related differential equations, it is obvious that the solution is :
                $$bar u_r(r)=c_1 I_k(r)+c_2 K_k(r)$$
                The functions $I_k(r)$ and $K_k(r)$ are the Modified Bessel functions of first and second kind respectively.



                http://mathworld.wolfram.com/ModifiedBesselDifferentialEquation.html



                http://mathworld.wolfram.com/ModifiedBesselFunctionoftheSecondKind.html



                http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html



                There is nothing to prove since this is the direct consequence of the definition of the Bessel functions and of the basic properties.



                Of course, $bar u_r(r)=0$ is a particular solution which satisfies the ODE (case $c_1=c_2=0$ ).



                In the above general solution $c_1$ and $c_2$ are arbitrary constants. Those constants can be determined if some well posed boundary conditions are specified.



                If in trying to solve the PDE you end up with the zero solution only, several causes can be suspected. A possible one could be that you introduces some "hided" or "implicit" conditions which are not in the wording of the problem. All depends on what method of solving you use and what exactly is your calculus. Without checking all steps of your calculus and without seeing all your intermediate equations it is impossible to give a definitive diagnosis.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 16 at 16:38









                JJacquelinJJacquelin

                45.4k21856




                45.4k21856






























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