Solutions to Differential Equations: Special Functions 1 Visualizing a few special functions 2 Lecture-26.nb Bessel's equation ‡ Bessel's equation is a second-order ODE having the following form, with the parameter n being real and nonnegative. In[155]:= 1 BesselODE = x2 y'' x + x y' x + x2 - n2 y x ã 0 @ D @ D H L @ D Lecture-26.nb 3 Out[155]= x2 - n2 y x + x y¢ x + x2 y¢¢ x ä 0 Bessel'sI equationM @ arisesD in@ problemsD @in Dheat conduction and mass diffusion for prob- lems with cylindrical symmetry (and many other situations). It's solution is a linear combination of two special functions called Bessel functions of the first kind, BesselJ in Mathematica, and Bessel functions of the second kind, BesselY in Mathematica. 4 Lecture-26.nb In[156]:= 2 DSolve BesselODE, y x , x Out[156]= @ y x Æ BesselJ@ D n,D x C 1 + BesselY n, x C 2 88 @ D @ D @ D @ D @ D<< Lecture-26.nb 5 ‡ The functions BesselJ and BesselY can be evaluated numerically and of course Mathematica has this capability built-in. In[157]:= MyPlotStyle HowMany_Integer := 3 Table Hue 0.66* i HowMany , Thickness 0.01 , i, 0, HowMany In[158]:= @ D 4 Plot BesselJ 0, x , BesselY 0, x , x, 0, 20 , PlotStyle Ø MyPlotStyle 2 @8 @ ê D @ D< 8 <D @8 @ 1D @ D< 8 < @ DD 0.5 5 10 15 20 -0.5 -1 6 Lecture-26.nb 1 0.5 5 10 15 20 -0.5 -1 1 0.5 Lecture-26.nb 7 5 10 15 20 -0.5 -1 1 0.5 5 10 15 20 8 -0.5 Lecture-26.nb -1 Out[158]= Ö Graphics Ö In[159]:= 5 Plot BesselJ 5, x , BesselY 5, x , x, 0, 20 , PlotStyle Ø MyPlotStyle 2 @8 @ D @ D< 8 < @ DD 5 10 15 20 -1 -2 -3 -4 -5 Lecture-26.nb 9 5 10 15 20 -1 -2 -3 -4 -5 5 10 15 20 -1 10 Lecture-26.nb -2 -3 -4 -5 5 10 15 20 -1 -2 -3 -4 Lecture-26.nb 11 -5 Out[159]= Ö Graphics Ö In[160]:= 6 Plot BesselJ 1 2, x , BesselY 1 2, x , x, 0, 20 , PlotStyle Ø MyPlotStyle 2 @8 @ ê D @ ê D< 8 < @ DD 0.5 0.25 5 10 15 20 -0.25 -0.5 -0.75 -1 12 Lecture-26.nb 0.5 0.25 5 10 15 20 -0.25 -0.5 -0.75 -1 0.5 0.25 Lecture-26.nb 13 5 10 15 20 -0.25 -0.5 -0.75 -1 0.5 0.25 5 10 15 20 -0.25 -0.5 14 Lecture-26.nb -0.75 -1 Out[160]= Ö Graphics Ö As an example of an application where the Bessel functions arise, consider radially-sym- metric heat flow in an infinitely long cylinder of radius a that has an initial temperature -ka2 t profile T =T(r) and a surface temperature T(r = a) = Tr , and substitute T = ‰ t into ∂T the heat equation ÅÅÅÅÅÅÅÅÅÅ = k “2 T in cylindrical coordinates. The function t must ∂t d2 t 1 dt obey ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ + a2 t = 0, which has the form of Bessel's equation of order n = ∂r2 r dr -ka2 t zero. The solution will have the form T = A J0 ar ‰ where A is a constant and J0 ar is the Bessel function of order zero of the first kind. The parameter a must, from boundary conditions, be a solution to the equation J(aa) = 0. The initial temperature distribution, f(r), can be expanded as a series Hof BesselL functions of zero order, analo- gousH toL what we did with Fourier series. The general solution to the heat-flow problem will then be an infinite series of Bessel functions, each modified by an exponentially ¶ 2 -kan t decaying amplitude, taking the form T = An J0 an r ‰ . For a more thorough n=1 discussion, see H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Second Edition, pp. 194–196‚ (1959).H L Lecture-26.nb 15 As an example of an application where the Bessel functions arise, consider radially-sym- metric heat flow in an infinitely long cylinder of radius a that has an initial temperature -ka2 t profile T =T(r) and a surface temperature T(r = a) = Tr , and substitute T = ‰ t into ∂T the heat equation ÅÅÅÅÅÅÅÅÅÅ = k “2 T in cylindrical coordinates. The function t must ∂t d2 t 1 dt obey ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ + a2 t = 0, which has the form of Bessel's equation of order n = ∂r2 r dr -ka2 t zero. The solution will have the form T = A J0 ar ‰ where A is a constant and J0 ar is the Bessel function of order zero of the first kind. The parameter a must, from boundary conditions, be a solution to the equation J(aa) = 0. The initial temperature distribution, f(r), can be expanded as a series Hof BesselL functions of zero order, analo- gousH toL what we did with Fourier series. The general solution to the heat-flow problem will then be an infinite series of Bessel functions, each modified by an exponentially ¶ 2 -kan t decaying amplitude, taking the form T = An J0 an r ‰ . For a more thorough n=1 discussion, see H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Second Edition, pp. 194–196‚ (1959).H L As an example of an application where the Bessel functions arise, consider radially-sym- metric heat flow in an infinitely long cylinder of radius a that has an initial temperature -ka2 t profile T =T(r) and a surface temperature T(r = a) = Tr , and substitute T = ‰ t into ∂T the heat equation ÅÅÅÅÅÅÅÅÅÅ = k “2 T in cylindrical coordinates. The function t must ∂t d2 t 1 dt obey ÅÅÅÅÅÅÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅ + a2 t = 0, which has the form of Bessel's equation of order n = ∂r2 r dr -ka2 t zero. The solution will have the form T = A J0 ar ‰ where A is a constant and J0 ar is the Bessel function of order zero of the first kind. The parameter a must, from boundary conditions, be a solution to the equation J(aa) = 0. The initial temperature 16 distribution, f(r), can be expanded as a series Hof BesselL functions of zero order,Lecture-26.nb analo- gousH toL what we did with Fourier series. The general solution to the heat-flow problem will then be an infinite series of Bessel functions, each modified by an exponentially ¶ 2 -kan t decaying amplitude, taking the form T = An J0 an r ‰ . For a more thorough n=1 discussion, see H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Second Edition, pp. 194–196‚ (1959).H L Legendre's equation Legendre's equation is a second-order ODE having the following form, with the parame- ters m and n being integers such that n is positive and -n ≤ m ≤ n (seems to suggest applications to quantum numbers!). Legendre's equation arises in physical problems with spherical symmetry. The simplest form of the equation has only one parameter, n and takes the form: Lecture-26.nbLegendre's equation is a second-order ODE having the following form, with the parame-17 ters m and n being integers such that n is positive and -n ≤ m ≤ n (seems to suggest applications to quantum numbers!). Legendre's equation arises in physical problems with spherical symmetry. The simplest form of the equation has only one parameter, n and takes the form: In[161]:= 7 LegendreODE = 1 - x2 y'' x - 2 x y' x + n n + 1 y x ã 0 Out[161]= 12 y x -H2 x y¢ Lx +@ D1 - x2 y@¢¢ Dx äH 0H LL @ D @ D @ D I M @ D 18 Lecture-26.nb ‡ Its solution is a linear combination of two special functions, Legendre and PLegendreQ: In[162]:= 8 DSolve LegendreODE, y x , x @ @ D D Lecture-26.nb 19 Out[162]= 1 y x Æ ÄÄÄÄÄÄ -3 x + 5 x3 C 1 + 2 2 5 x2 1 1 1 C 2 ÄÄÄÄÄÄ - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ - ÄÄÄÄÄÄ x 3 - 5 x2 - ÄÄÄÄÄ Log 1 - x + ÄÄÄÄÄ Log 1 + x 99 @ D I 3 2 M @2D 2 2 i y j ji zyz @ D j I M j @ D @ Dzz== k k {{ 20 Lecture-26.nb ‡ The two-parameter form of the Legendre equation is: In[163]:= m2 9 AnotherFormLegendreODE = 1 - x2 y'' x - 2 x y' x + n n + 1 - ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ y x ã 0 1 - x2 Out[163]= i y ¢ H 2 L ¢¢ @ D @ D j H L z @ D 12 y x - 2 x y x + 1 - x y x ä 0 k { @ D @ D I M @ D Lecture-26.nb 21 ‡ which has a solution involving the same special functions: In[164]:= 10 DSolve AnotherFormLegendreODE, y x , x @ @ D D 22 Lecture-26.nb Out[164]= 1 y x Æ ÄÄÄÄÄÄ -3 x + 5 x3 C 1 + 2 2 5 x2 1 1 1 C 2 ÄÄÄÄÄÄ - ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ - ÄÄÄÄÄÄ x 3 - 5 x2 - ÄÄÄÄÄ Log 1 - x + ÄÄÄÄÄ Log 1 + x 99 @ D I 3 2 M @2D 2 2 i y j ji zyz @ D j I M j @ D @ Dzz== k k {{ Lecture-26.nb 23 ‡ Of course, Mathematica can evaluate and plot Legendre functions… In[165]:= 11 Plot LegendreP 0, x , LegendreQ 0, x , x, -1, 1 , PlotStyle Ø MyPlotStyle 2 @ D @8 @ D @ D< 8 < 7.5 D 5 2.5 -1 -0.5 0.5 1 -2.5 -5 -7.5 24 7.5 Lecture-26.nb 5 2.5 -1 -0.5 0.5 1 -2.5 -5 -7.5 7.5 5 2.5 Lecture-26.nb 25 -1 -0.5 0.5 1 -2.5 -5 -7.5 7.5 5 2.5 -1 -0.5 0.5 1 -2.5 -5 26 -7.5 Lecture-26.nb Out[165]= Ö Graphics Ö In[166]:= 12 Plot LegendreP 1, x , LegendreQ 1, x , x, -1, 1 , PlotStyle Ø MyPlotStyle 2 4 @8 @ D @ D< 8 < @ DD 3 2 1 -1 -0.5 0.5 1 -1 4 Lecture-26.nb 27 3 2 1 -1 -0.5 0.5 1 -1 4 3 2 28 Lecture-26.nb 1 -1 -0.5 0.5 1 -1 Lecture-26.nb 29 Out[166]= Ö Graphics Ö In[167]:= 13 Plot LegendreP 2, x , LegendreQ 2, x , x, -1, 1 , PlotStyle Ø MyPlotStyle 2 @8 @ D @ D< 8 < @ DD 3 2 1 -1 -0.5 0.5 1 -1 -2 -3 30 3 Lecture-26.nb 2 1 -1 -0.5 0.5 1 -1 -2 -3 3 2 1 Lecture-26.nb 31 -1 -0.5 0.5 1 -1 -2 -3 32 Lecture-26.nb Out[167]= Ö Graphics Ö In[168]:= 14 Plot Evaluate Table LegendreP i, x , i, 0, 10 , x, -1, 1 , PlotStyle Ø MyPlotStyle 11 1 @ @ @ @ D 8 <DD 8 < @ DD 0.5 -1 -0.5 0.5 1 -0.5 -1 1 Lecture-26.nb 33 0.5 -1 -0.5 0.5 1 -0.5 -1 1 0.5 34 Lecture-26.nb -1 -0.5 0.5 1 -0.5 -1 Lecture-26.nb 35 Out[168]= Ö Graphics Ö In[169]:= 15 Plot Evaluate Table LegendreQ i, x , i, 0, 10 , x, -1, 1 , PlotStyle Ø MyPlotStyle 11 @ @ @ @ D 8 <DD 8 < @ DD 2 1 -1 -0.5 0.5 1 -1 -2 36 2 Lecture-26.nb 1 -1 -0.5 0.5 1 -1 -2 2 1 Lecture-26.nb 37 -1 -0.5 0.5 1 -1 -2 38 Lecture-26.nb Out[169]= Ö Graphics Ö Lecture-26.nb 39 ‡ Note how some of the Legendre functions are even and some are odd.