Solving Fredholm Equation of the second kind












6












$begingroup$


Consider the Fredholm Equation of the second kind,



$$phi(x) = 3 + lambda int_{0}^{pi} text{cos}(x-s) , phi(s) ,ds$$



Where the analytical solution is found as,



$$phi(x) = 3 + frac{6lambda}{1 - lambda frac{pi}{2}},text{sin}(x)$$



How could one use Mathematica to find a numerical solution to the same integral equation by using the method of successive approximations (i.e. the Neumann series approach)?






numerical-integration integral-equations







share|improve this question











$endgroup$

















    6












    $begingroup$


    Consider the Fredholm Equation of the second kind,



    $$phi(x) = 3 + lambda int_{0}^{pi} text{cos}(x-s) , phi(s) ,ds$$



    Where the analytical solution is found as,



    $$phi(x) = 3 + frac{6lambda}{1 - lambda frac{pi}{2}},text{sin}(x)$$



    How could one use Mathematica to find a numerical solution to the same integral equation by using the method of successive approximations (i.e. the Neumann series approach)?






    numerical-integration integral-equations







    share|improve this question











    $endgroup$















      6












      6








      6





      $begingroup$


      Consider the Fredholm Equation of the second kind,



      $$phi(x) = 3 + lambda int_{0}^{pi} text{cos}(x-s) , phi(s) ,ds$$



      Where the analytical solution is found as,



      $$phi(x) = 3 + frac{6lambda}{1 - lambda frac{pi}{2}},text{sin}(x)$$



      How could one use Mathematica to find a numerical solution to the same integral equation by using the method of successive approximations (i.e. the Neumann series approach)?






      numerical-integration integral-equations







      share|improve this question











      $endgroup$




      Consider the Fredholm Equation of the second kind,



      $$phi(x) = 3 + lambda int_{0}^{pi} text{cos}(x-s) , phi(s) ,ds$$



      Where the analytical solution is found as,



      $$phi(x) = 3 + frac{6lambda}{1 - lambda frac{pi}{2}},text{sin}(x)$$



      How could one use Mathematica to find a numerical solution to the same integral equation by using the method of successive approximations (i.e. the Neumann series approach)?






      numerical-integration integral-equations




      numerical-integration integral-equations






      share|improve this question















      share|improve this question













      share|improve this question




      share|improve this question








      edited Mar 2 at 20:25









      J. M. is computer-less♦

      97.3k10303463




      97.3k10303463










      asked Mar 1 at 19:28









      LightningStrikeLightningStrike

      735




      735






















          2 Answers
          2






          active

          oldest

          votes


















          8












          $begingroup$

          Use DSolve:



          PHI = 
          
  DSolveValue[ϕ[x] == 3 + λ Integrate[ Cos[x - s] ϕ[s], {s, 0, Pi}], ϕ, x]
          
(*Function[{x}, (3 (-2 + π λ - 4 λ Sin[x]))/(-2 + π λ)]*)


          The solution can be further used in the form PHI[x].






          share|improve this answer











          $endgroup$













          • $begingroup$
            Thank you, but how can I use the function Mathematica returns to, say, investigate the convergence of the new $phi (x)$ function?
            $endgroup$
            – LightningStrike
            Mar 1 at 19:40






          • 1




            $begingroup$
            @ user57401 I modified my answer!
            $endgroup$
            – Ulrich Neumann
            Mar 1 at 19:57



















          3












          $begingroup$

          Following Weisstein, Eric W. "Integral Equation Neumann Series." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html, the Neumann series approximation is:



          n = 10;  (* for example *)
          
ϕ[x_, 0] = 3;
          
Do[ϕ[x_, j_] = 3 + λ Integrate[Cos[x - p] ϕ[p, j - 1], {p, 0, π}], {j, n}]


          The last term in the series ϕ[x,n] is the approximation to ϕ[x].



          Here is what Mathematica returns for ϕ[x,10].



          phi_of_ten



          To investigate convergence, I guess we could look at the difference ϕ[x,n] - ϕ[x] as n gets large, since you know ϕ[x].






          share|improve this answer











          $endgroup$













          • $begingroup$
            Thank you! When I try to run this, my output is returning the value of 3? How did you get Mathematica to return the series above for [Phi][10]?
            $endgroup$
            – LightningStrike
            Mar 1 at 20:27










          • $begingroup$
            Please clear out your variables, perhaps with Evaluation: Quit Kernel: Local. To print the final (nth) value: [Phi][x, n].
            $endgroup$
            – mjw
            Mar 1 at 20:52












          • $begingroup$
            Made some edits to my answer. Had a couple of typos. Within a function definition it is x_, otherwise x. Also, [Phi][x,j] needs two arguments, one for x and one for the jth approximation. Hope its clear.
            $endgroup$
            – mjw
            Mar 1 at 20:55












          • $begingroup$
            @m_goldberg, how do you post symbols rather than the clutzy [Phi] type of notation here?
            $endgroup$
            – mjw
            Mar 3 at 1:57






          • 1




            $begingroup$
            I use halirutan's plug-in. You can learn more about it here
            $endgroup$
            – m_goldberg
            Mar 3 at 2:53











          Your Answer





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          2 Answers
          2






          active

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          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          8












          $begingroup$

          Use DSolve:



          PHI = 
          
  DSolveValue[ϕ[x] == 3 + λ Integrate[ Cos[x - s] ϕ[s], {s, 0, Pi}], ϕ, x]
          
(*Function[{x}, (3 (-2 + π λ - 4 λ Sin[x]))/(-2 + π λ)]*)


          The solution can be further used in the form PHI[x].






          share|improve this answer











          $endgroup$













          • $begingroup$
            Thank you, but how can I use the function Mathematica returns to, say, investigate the convergence of the new $phi (x)$ function?
            $endgroup$
            – LightningStrike
            Mar 1 at 19:40






          • 1




            $begingroup$
            @ user57401 I modified my answer!
            $endgroup$
            – Ulrich Neumann
            Mar 1 at 19:57
















          8












          $begingroup$

          Use DSolve:



          PHI = 
          
  DSolveValue[ϕ[x] == 3 + λ Integrate[ Cos[x - s] ϕ[s], {s, 0, Pi}], ϕ, x]
          
(*Function[{x}, (3 (-2 + π λ - 4 λ Sin[x]))/(-2 + π λ)]*)


          The solution can be further used in the form PHI[x].






          share|improve this answer











          $endgroup$













          • $begingroup$
            Thank you, but how can I use the function Mathematica returns to, say, investigate the convergence of the new $phi (x)$ function?
            $endgroup$
            – LightningStrike
            Mar 1 at 19:40






          • 1




            $begingroup$
            @ user57401 I modified my answer!
            $endgroup$
            – Ulrich Neumann
            Mar 1 at 19:57














          8












          8








          8





          $begingroup$

          Use DSolve:



          PHI = 
          
  DSolveValue[ϕ[x] == 3 + λ Integrate[ Cos[x - s] ϕ[s], {s, 0, Pi}], ϕ, x]
          
(*Function[{x}, (3 (-2 + π λ - 4 λ Sin[x]))/(-2 + π λ)]*)


          The solution can be further used in the form PHI[x].






          share|improve this answer











          $endgroup$



          Use DSolve:



          PHI = 
          
  DSolveValue[ϕ[x] == 3 + λ Integrate[ Cos[x - s] ϕ[s], {s, 0, Pi}], ϕ, x]
          
(*Function[{x}, (3 (-2 + π λ - 4 λ Sin[x]))/(-2 + π λ)]*)


          The solution can be further used in the form PHI[x].







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited Mar 2 at 23:50









          m_goldberg

          87.4k872198




          87.4k872198










          answered Mar 1 at 19:34









          Ulrich NeumannUlrich Neumann

          9,513616




          9,513616












          • $begingroup$
            Thank you, but how can I use the function Mathematica returns to, say, investigate the convergence of the new $phi (x)$ function?
            $endgroup$
            – LightningStrike
            Mar 1 at 19:40






          • 1




            $begingroup$
            @ user57401 I modified my answer!
            $endgroup$
            – Ulrich Neumann
            Mar 1 at 19:57


















          • $begingroup$
            Thank you, but how can I use the function Mathematica returns to, say, investigate the convergence of the new $phi (x)$ function?
            $endgroup$
            – LightningStrike
            Mar 1 at 19:40






          • 1




            $begingroup$
            @ user57401 I modified my answer!
            $endgroup$
            – Ulrich Neumann
            Mar 1 at 19:57
















          $begingroup$
          Thank you, but how can I use the function Mathematica returns to, say, investigate the convergence of the new $phi (x)$ function?
          $endgroup$
          – LightningStrike
          Mar 1 at 19:40




          $begingroup$
          Thank you, but how can I use the function Mathematica returns to, say, investigate the convergence of the new $phi (x)$ function?
          $endgroup$
          – LightningStrike
          Mar 1 at 19:40




          1




          1




          $begingroup$
          @ user57401 I modified my answer!
          $endgroup$
          – Ulrich Neumann
          Mar 1 at 19:57




          $begingroup$
          @ user57401 I modified my answer!
          $endgroup$
          – Ulrich Neumann
          Mar 1 at 19:57











          3












          $begingroup$

          Following Weisstein, Eric W. "Integral Equation Neumann Series." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html, the Neumann series approximation is:



          n = 10;  (* for example *)
          
ϕ[x_, 0] = 3;
          
Do[ϕ[x_, j_] = 3 + λ Integrate[Cos[x - p] ϕ[p, j - 1], {p, 0, π}], {j, n}]


          The last term in the series ϕ[x,n] is the approximation to ϕ[x].



          Here is what Mathematica returns for ϕ[x,10].



          phi_of_ten



          To investigate convergence, I guess we could look at the difference ϕ[x,n] - ϕ[x] as n gets large, since you know ϕ[x].






          share|improve this answer











          $endgroup$













          • $begingroup$
            Thank you! When I try to run this, my output is returning the value of 3? How did you get Mathematica to return the series above for [Phi][10]?
            $endgroup$
            – LightningStrike
            Mar 1 at 20:27










          • $begingroup$
            Please clear out your variables, perhaps with Evaluation: Quit Kernel: Local. To print the final (nth) value: [Phi][x, n].
            $endgroup$
            – mjw
            Mar 1 at 20:52












          • $begingroup$
            Made some edits to my answer. Had a couple of typos. Within a function definition it is x_, otherwise x. Also, [Phi][x,j] needs two arguments, one for x and one for the jth approximation. Hope its clear.
            $endgroup$
            – mjw
            Mar 1 at 20:55












          • $begingroup$
            @m_goldberg, how do you post symbols rather than the clutzy [Phi] type of notation here?
            $endgroup$
            – mjw
            Mar 3 at 1:57






          • 1




            $begingroup$
            I use halirutan's plug-in. You can learn more about it here
            $endgroup$
            – m_goldberg
            Mar 3 at 2:53
















          3












          $begingroup$

          Following Weisstein, Eric W. "Integral Equation Neumann Series." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html, the Neumann series approximation is:



          n = 10;  (* for example *)
          
ϕ[x_, 0] = 3;
          
Do[ϕ[x_, j_] = 3 + λ Integrate[Cos[x - p] ϕ[p, j - 1], {p, 0, π}], {j, n}]


          The last term in the series ϕ[x,n] is the approximation to ϕ[x].



          Here is what Mathematica returns for ϕ[x,10].



          phi_of_ten



          To investigate convergence, I guess we could look at the difference ϕ[x,n] - ϕ[x] as n gets large, since you know ϕ[x].






          share|improve this answer











          $endgroup$













          • $begingroup$
            Thank you! When I try to run this, my output is returning the value of 3? How did you get Mathematica to return the series above for [Phi][10]?
            $endgroup$
            – LightningStrike
            Mar 1 at 20:27










          • $begingroup$
            Please clear out your variables, perhaps with Evaluation: Quit Kernel: Local. To print the final (nth) value: [Phi][x, n].
            $endgroup$
            – mjw
            Mar 1 at 20:52












          • $begingroup$
            Made some edits to my answer. Had a couple of typos. Within a function definition it is x_, otherwise x. Also, [Phi][x,j] needs two arguments, one for x and one for the jth approximation. Hope its clear.
            $endgroup$
            – mjw
            Mar 1 at 20:55












          • $begingroup$
            @m_goldberg, how do you post symbols rather than the clutzy [Phi] type of notation here?
            $endgroup$
            – mjw
            Mar 3 at 1:57






          • 1




            $begingroup$
            I use halirutan's plug-in. You can learn more about it here
            $endgroup$
            – m_goldberg
            Mar 3 at 2:53














          3












          3








          3





          $begingroup$

          Following Weisstein, Eric W. "Integral Equation Neumann Series." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html, the Neumann series approximation is:



          n = 10;  (* for example *)
          
ϕ[x_, 0] = 3;
          
Do[ϕ[x_, j_] = 3 + λ Integrate[Cos[x - p] ϕ[p, j - 1], {p, 0, π}], {j, n}]


          The last term in the series ϕ[x,n] is the approximation to ϕ[x].



          Here is what Mathematica returns for ϕ[x,10].



          phi_of_ten



          To investigate convergence, I guess we could look at the difference ϕ[x,n] - ϕ[x] as n gets large, since you know ϕ[x].






          share|improve this answer











          $endgroup$



          Following Weisstein, Eric W. "Integral Equation Neumann Series." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html, the Neumann series approximation is:



          n = 10;  (* for example *)
          
ϕ[x_, 0] = 3;
          
Do[ϕ[x_, j_] = 3 + λ Integrate[Cos[x - p] ϕ[p, j - 1], {p, 0, π}], {j, n}]


          The last term in the series ϕ[x,n] is the approximation to ϕ[x].



          Here is what Mathematica returns for ϕ[x,10].



          phi_of_ten



          To investigate convergence, I guess we could look at the difference ϕ[x,n] - ϕ[x] as n gets large, since you know ϕ[x].







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited yesterday

























          answered Mar 1 at 20:12









          mjwmjw

          5429




          5429












          • $begingroup$
            Thank you! When I try to run this, my output is returning the value of 3? How did you get Mathematica to return the series above for [Phi][10]?
            $endgroup$
            – LightningStrike
            Mar 1 at 20:27










          • $begingroup$
            Please clear out your variables, perhaps with Evaluation: Quit Kernel: Local. To print the final (nth) value: [Phi][x, n].
            $endgroup$
            – mjw
            Mar 1 at 20:52












          • $begingroup$
            Made some edits to my answer. Had a couple of typos. Within a function definition it is x_, otherwise x. Also, [Phi][x,j] needs two arguments, one for x and one for the jth approximation. Hope its clear.
            $endgroup$
            – mjw
            Mar 1 at 20:55












          • $begingroup$
            @m_goldberg, how do you post symbols rather than the clutzy [Phi] type of notation here?
            $endgroup$
            – mjw
            Mar 3 at 1:57






          • 1




            $begingroup$
            I use halirutan's plug-in. You can learn more about it here
            $endgroup$
            – m_goldberg
            Mar 3 at 2:53


















          • $begingroup$
            Thank you! When I try to run this, my output is returning the value of 3? How did you get Mathematica to return the series above for [Phi][10]?
            $endgroup$
            – LightningStrike
            Mar 1 at 20:27










          • $begingroup$
            Please clear out your variables, perhaps with Evaluation: Quit Kernel: Local. To print the final (nth) value: [Phi][x, n].
            $endgroup$
            – mjw
            Mar 1 at 20:52












          • $begingroup$
            Made some edits to my answer. Had a couple of typos. Within a function definition it is x_, otherwise x. Also, [Phi][x,j] needs two arguments, one for x and one for the jth approximation. Hope its clear.
            $endgroup$
            – mjw
            Mar 1 at 20:55












          • $begingroup$
            @m_goldberg, how do you post symbols rather than the clutzy [Phi] type of notation here?
            $endgroup$
            – mjw
            Mar 3 at 1:57






          • 1




            $begingroup$
            I use halirutan's plug-in. You can learn more about it here
            $endgroup$
            – m_goldberg
            Mar 3 at 2:53
















          $begingroup$
          Thank you! When I try to run this, my output is returning the value of 3? How did you get Mathematica to return the series above for [Phi][10]?
          $endgroup$
          – LightningStrike
          Mar 1 at 20:27




          $begingroup$
          Thank you! When I try to run this, my output is returning the value of 3? How did you get Mathematica to return the series above for [Phi][10]?
          $endgroup$
          – LightningStrike
          Mar 1 at 20:27












          $begingroup$
          Please clear out your variables, perhaps with Evaluation: Quit Kernel: Local. To print the final (nth) value: [Phi][x, n].
          $endgroup$
          – mjw
          Mar 1 at 20:52






          $begingroup$
          Please clear out your variables, perhaps with Evaluation: Quit Kernel: Local. To print the final (nth) value: [Phi][x, n].
          $endgroup$
          – mjw
          Mar 1 at 20:52














          $begingroup$
          Made some edits to my answer. Had a couple of typos. Within a function definition it is x_, otherwise x. Also, [Phi][x,j] needs two arguments, one for x and one for the jth approximation. Hope its clear.
          $endgroup$
          – mjw
          Mar 1 at 20:55






          $begingroup$
          Made some edits to my answer. Had a couple of typos. Within a function definition it is x_, otherwise x. Also, [Phi][x,j] needs two arguments, one for x and one for the jth approximation. Hope its clear.
          $endgroup$
          – mjw
          Mar 1 at 20:55














          $begingroup$
          @m_goldberg, how do you post symbols rather than the clutzy [Phi] type of notation here?
          $endgroup$
          – mjw
          Mar 3 at 1:57




          $begingroup$
          @m_goldberg, how do you post symbols rather than the clutzy [Phi] type of notation here?
          $endgroup$
          – mjw
          Mar 3 at 1:57




          1




          1




          $begingroup$
          I use halirutan's plug-in. You can learn more about it here
          $endgroup$
          – m_goldberg
          Mar 3 at 2:53




          $begingroup$
          I use halirutan's plug-in. You can learn more about it here
          $endgroup$
          – m_goldberg
          Mar 3 at 2:53


















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