Bessel-Function

• This repository contains codes for plotting and visualizing Bessel functions and its applications -

Several second order ODEs of the form 𝒚 ′′ + 𝒑(𝒙)𝒚 ′ + 𝒒(𝒙)𝒚 = 𝒓(𝒙) are of practical importance have Power series solution

if coefficients p(x), q(x) and r(x) are functions instead of constant coefficients. Further, if they must have valid Taylor series expansion about point 𝑥0, means they must be continuously differentiable about that point i.e. they are analytical at that point. If the coefficients p(x), q(x), r(x) are not analytical at point 𝑥0 but if we still require a power series solution at that point, in order to exploit the larger radius of convergence, we use Frobenius method. Frobenius methods masks the point of singularity, thereby creating feasible solution at which the power series method fails. Such points are called regular singular points. Consider an example ODE:

In above example problem p(𝑥) and q(𝑥) are undefined at 𝑥 = 0 but we can still apply frobenius method if 𝑥0 is regular singular point of ODE. The solution according to Frobenius is by

𝑥0 is the regular singular point of

if (𝑥 − 𝑥0 )𝑝(𝑥) and (𝑥 − 𝑥0 ) 2𝑞(𝑥) exist and has valid Taylor expansion about 𝑥0. The exponent r (may be real or complex) number should be chosen such that 𝑎0 ≠ 0. Now, there exists a class of 2 nd order, linear ODEs with variable coefficients of the form:

The Bessel function of the first kind of mth order is given by:

J_m(x) =

The behaviour of the Bessel functions of first kind 𝐽𝑚 of order ‘m’ are shown below:

The behaviour of the Bessel functions of second kind 𝑌𝑚 of order ‘m’ are shown below:

A general solution of Bessel’s function for the Bessel ODE is given by 𝑦(𝑥) = 𝐶1𝐽_𝑚 + 𝐶2𝑌_m

APPLICATION 1: CYLINDER WITH ENERGY GENERATION

A long solid cylinder of radius ro is initially at uniform temperature Ti. Electricity is suddenly passed through the cylinder resulting in volumetric heat generation rate of qm. The cylinder is cooled by convection at its surface. The heat transfer coefficient is considered as h and the ambient temperature is considered as T∞. The objective is to determine the transient temperature of the cylinder.

Assumptions:

1. One dimensional conduction.
2. Uniform h and T∞.
3. Constant conductivity.
4. Constant diffusivity.
5. Negligible end effect.

Governing Equations: To make the convection boundary condition homogeneous, we introduce the following temperature variable θ (r,t) = T(r,t) - T∞.

Based on the above assumptions, gives

Boundary and initial conditions:

Solution: Since the differential equation s non-homogeneous, we assume a solution of the form 𝜃(𝑟,𝑡) = 𝜑(𝑟,𝑡) + ∅(𝑟) (a)

Note that Ψ(r-t) depends on two variables while ϕ(r) depends on one variable. Substituting (a) into eq. (A)

(b)

The next step is to split (b) into two equations, one for Ψ(r-t) and the other for ϕ(r). Let..

(c) (d)

To solve equations (c) and (d) we need two boundary conditions for each and an initial condition for (c). Substituting (a) into boundary condition (1)

Similarly, condition (2) gives

Now, the initial condition gives 𝜑(𝑟, 0) = (𝑇𝑖 − 𝑇∞) − ∅(𝑟) (c-3)

Integrating (d) gives ∅(𝑟) = −(𝑞_𝑚/4𝑘)*𝑟^2 + 𝑐1𝑙𝑛𝑟 + 𝑐2 (e)

Equation (c) is solved by the method of separation of variables. Assume a product solution of the form 𝜑(𝑟,𝑡) = 𝑅(𝑟)𝜏(𝑡) (f) Substituting (f) into (c), separating variables, and setting the resulting equation equal to a constant, ±(λ_k)^2, gives

ince the r-variable has two homogeneous conditions, the plus sign must be selected in (g). Equations (g) and (h) become

Solutions to the ODE (i) is given by general Bessel representation and (ii) is exponential decay: 𝑅_𝑘(𝑟) = (𝐴_𝑘)*𝐽0 (𝜆_𝑘)*r + (𝐵_𝑘)𝑌0(𝜆_𝑘)*r (k)

and 𝜏_𝑘(𝑡) = (𝐶_𝑘)exp(−𝛼𝑡(𝜆_𝑘)^2) (l)

Application to boundary and initial conditions. Conditions (c-1) and (c-2) give 𝐵_𝑘 = 0 and Bi𝐽0(𝜆_𝑘)𝑟0 = (𝜆_𝑘)𝑟0𝐽1*(𝜆_𝑘)*𝑟0 (m) Where Bi is the Biot number defined as Bi = hr0 / k. The roots of (m) give the constants λk. Substituting (k) and (l) into (f) and summing all solutions

Application of the non-homogeneous initial condition (c-3) yields

The characteristic functions J0 ((λ_k)*r) in equation (p) are solutions to (i). This is a Sturm-Liouville equation that guarantees that there are infinitely many eigen values and the function is orthogonal when the boundary conditions are homogeneous w.r.t. weight function w(r) = r. Multiplying both sides of (p) by J0((λ_k)*r)*r dr, integrating from r =0 to r =r0 and invoking orthogonality gives a_k

Substituting (o) into (q), and evaluate the integral gives,

Complete Solution: Hence the complete solution, expressed in dimensionless form, is