What is Bessel equation: Definition and 24 Discussions

Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation




















{\displaystyle x^{2}{\frac {d^{2}y}{dx^{2}}}+x{\frac {dy}{dx}}+\left(x^{2}-\alpha ^{2}\right)y=0}
for an arbitrary complex number α, the order of the Bessel function. Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of α.
The most important cases are when α is an integer or half-integer. Bessel functions for integer α are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer α are obtained when the Helmholtz equation is solved in spherical coordinates.

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  1. T

    I Modified Bessel Equation

    Hey all, I wanted to know if anyone knew somewhere I could find the asymptotic behavior for small x (i.e x approaching 0) limit of the modified Bessel equations with complex order. The wikipedia page for Bessel functions...
  2. I

    Use a variable substitution to get into a Bessel equation form?

    Hello, For my homework I am supposed to get- into the form of a Bessel equation using variable substitution. I am just not sure what substitution to use. Thanks in advance.
  3. K

    MATLAB MATLAB - solving equation with Bessel function

    Hello, i am trying to solve this equation for x besselj(0,0.5*x)*bessely(0,4.5*x)-besselj(0,4.5*x)*bessely(0,0.5*x) ==0; I tried vpasolve, but it gave me answer x=0 only. fzero function didnt work, too. What function can solve this equation? Thanks
  4. dykuma

    Partial Differential equation, Temp in a Cylinder

    Homework Statement Homework Equations The Attempt at a Solution Because we are only looking at a cross section, I tried to reduce 5.3 down to just being a function of R and Theta. However I reasoned that there should be, based on this problem, no dependence on Theta either, so I figured I...
  5. P

    Changing Independent Variable in the Bessel Equation

    Homework Statement Given the bessel equation $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} -(1-x)y=0$$ show that when changing the variable to ##u = 2\sqrt{x}## the equation becomes $$u^2\frac{d^2y}{du^2}+u\frac{dy}{du}+(u^2-4)y = 0$$ Homework Equations The Attempt at a Solution ##u=2\sqrt{x}##...
  6. K

    Oscillations of a free hanging chain

    Homework Statement I am trying to find an equation for a free hanging chain of mass m and length L. The chain is hanging vertically downwards where x is measured vertically upwards from the free end of the chain and y is measured horizontally. Homework Equations [/B] I derived this...
  7. Aleen Muhammed

    B Solving Naumann Equation: How Bessel's Second Kind Was Discovered

    How Naumann arrived to the second kind of Bessel ? I want proof , Please .. http://C:\Users\User\Downloads
  8. Aleen Muhammed

    Solve Derivative Bessel Function (Type II & III) - Help Needed

    Hello .. I have research on derivative Bessel type II and type III function (function Henkel), I can not get it .. Please help me.:cry:
  9. P

    Jackson Problem 3.12/3.18 -- Electric potential near two plates

    Homework Statement I need to solve a problem like Jackson 3.18. I need to find potential due to the same configuration but the position of two plates is opposite i.e. Plate at Z=0 contains disc with potential V and plate at Z=0 is grounded. Homework EquationsThe Attempt at a Solution I think...
  10. D

    Recurrence relations define solutions to Bessel equation

    I'm trying to show that a function defined with the following recurence relations $$\frac{dZ_m(x)}{dx}=\frac{1}{2}(Z_{m-1}-Z_{m+1})$$ and $$\frac{2m}{x}Z_m=Z_{m+1}+Z_{m-1}$$ satisfies the Bessel differential equation $$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$...
  11. U

    Bessel Equation and Bessel fuctions

    We first express Bessel's Equation in Sturm-Liouville form through a substitution: Next, we consider a series solution and replace v by m where m is an integer. We obtain a recurrence relation: Then, since all these terms must be = 0, Consider m = 0 First term vanishes and second term = a1x...
  12. M

    Legendre equation , the Bessel equation and Sturm Liouville equation

    Show that the Legendre equation as well as the Bessel equation for n=integer are Sturm Liouville equations and thus their solutions are orthogonal. How I can proove that ..? :(
  13. A

    Can Bessel's equation be solved using only Frobenius method?

    can we use only frobenius method to solve bessel equation?
  14. W

    Laplace Transform of a Bessel Equation

    Hi guys, I have this question on Laplace transforms, but am not sure how to start it. The zero order Bessel function Jo(t) satisfies the ordinary differential equation: tJ''o(t) + J'o(t) + tJo(t) = 0 Take the Laplace transform of this equation and use the properties of the transform to find...
  15. T

    Verifying the integral form of the Bessel equation by substitution

    Homework Statement The following is an integral form of the Bessel equation of order n: J_n(x) = \frac{1}{\pi}\int_0^{\pi}\ \cos(x\sin(t)-nt)\ dt Show by substitution that this satisfies the Bessel equation of order n. Homework Equations Bessel equation of order n: x^2y'' + xy' +...
  16. Telemachus

    Solving the Bessel Equation: Find Solutions & Justify

    Hi there. I'm working with the Bessel equation, and I have this problem. It says: a) Given the equation \frac{d^2y}{dt^2}+\frac{1}{t}\frac{dy}{dt}+4t^2y(t)=0 Use the substitution x=t^2 to find the general solution b) Find the particular solution that verifies y(0)=5 c) Does any solution...
  17. M

    Numerical evaluation of modified Bessel equation

    Not sure if this is the right place. Mathematica has a function BesselK[0,x] that returns the value of the modified Bessel function K_0 at x. Is there public documentation of how this algorithm works? If not, is there documentation regarding any algorithm of K_0? I am hoping it doesn't...
  18. H

    Problem in Bessel equation help .

    problem in Bessel equation help ... Homework Statement using the formula d\dx (x^n Jn(x))=x^n Jn-1(x) & 2n\x Jn(x)=Jn+1(x)+Jn-1(x) Homework Equations prove that integral from 0 to 1 (x(1-x^2)Jdot(x) dx = 4 J1(1) - 2 Jdot (1) The Attempt at a Solution it's difficult one i can not...
  19. D

    Converting a Differential Equation to Bessel Equation

    Hi all can anyone help me to reduce following diff.Equ. to bessel eq. 4x^3*y''-y=0 thanks in advance . I am also still trying to show that it can be converted to bessel function.
  20. T

    Bessel equation & Orthogonal Basis

    I remember some of my linear algebra from my studies but can't wrap my head around this one. Homework Statement Say my solution to a DE is "f(x)" (happens to be bessel's equation), and it contains a constant variable "d" in the argument of the bessel's functions (i.,e. J(d*x) and Y(d*x)). So...
  21. Y

    What did I do wrong in this Bessel equation?

    I need to convertx^{2}y''+2xy'+[kx^{2}-n(n+1)]y=0 using y=x^{-\frac{1}{2}}w to a normal Modified Bessel Equation and I cannot get to that. I check many times and I must be having a blind spot! This is my work: y=x^{-\frac{1}{2}} w \Rightarrow...
  22. Z

    A more general Bessel equation?

    In the solution to a recent problem set, my prof referenced a "general Bessel ODE" which he gave in the form: x^{2}\frac{d^{2}y}{dx^{2}}+x\left(a+2bx^{q}\right)\frac{dy}{dx}+\left[c+dx^{2s}-b\left(1-a-q\right)x^{q}+b^{2}x^{2q}\right]y=0 The only format of the Bessel ODE that appears in the...
  23. L

    Why Does n_0(x) Fail to Satisfy the Spherical Bessel Equation?

    What am I missing when I'm unsuccessful in showing by direct substitution into the spherical Bessel equation r^2 \frac{d^2R}{dr^2} + 2r \frac{dR}{dr} + [k^2 r^2 - n(n + 1)] R = 0 that n_0 (x) = - \frac{1}{x} \sum_{s \geq 0} \frac{(-1)^s}{(2s)!} x^{2s} is a solution? What's the catch??
  24. A

    Linear dependence of bessel equation

    why for bessel equations, if n isn't an integer, you can have the solution y(x)=(c1)Jn(x) +(c2)J(-n)x but isn't true if n's an integer?