- This repository contains codes for plotting and visualizing Bessel functions and its applications -
Several second order ODEs of the form π β²β² + π(π)π β² + π(π)π = π(π) 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 π₯0, 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 π₯0 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(π₯) and q(π₯) are undefined at π₯ = 0 but we can still apply frobenius method if π₯0 is regular singular point of ODE. The solution according to Frobenius is by
π₯0 is the regular singular point of
if (π₯ β π₯0 )π(π₯) and (π₯ β π₯0 ) 2π(π₯) exist and has valid Taylor expansion about π₯0. The exponent r (may be real or complex) number should be chosen such that π0 β 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:
The behaviour of the Bessel functions of first kind π½π of order βmβ are shown below:
The behaviour of the Bessel functions of second kind ππ of order βmβ are shown below:
A general solution of Besselβs function for the Bessel ODE is given by π¦(π₯) = πΆ1π½_π + πΆ2π_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 Tβ. The objective is to determine the transient temperature of the cylinder.
Assumptions:
- One dimensional conduction.
- Uniform h and Tβ.
- Constant conductivity.
- Constant diffusivity.
- Negligible end effect.
Governing Equations: To make the convection boundary condition homogeneous, we introduce the following temperature variable ΞΈ (r,t) = T(r,t) - Tβ.
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 π(π,π‘) = π(π,π‘) + β (π) (a)
Note that Ξ¨(r-t) depends on two variables while Ο(r) depends on one variable. Substituting (a) into eq. (A)
The next step is to split (b) into two equations, one for Ξ¨(r-t) and the other for Ο(r). Let..
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 π(π, 0) = (ππ β πβ) β β (π) (c-3)
Integrating (d) gives β (π) = β(π_π/4π)*π^2 + π1πππ + π2 (e)
Equation (c) is solved by the method of separation of variables. Assume a product solution of the form π(π,π‘) = π (π)π(π‘) (f) Substituting (f) into (c), separating variables, and setting the resulting equation equal to a constant, Β±(Ξ»_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: π _π(π) = (π΄_π)*π½0 (π_π)*r + (π΅_π)π0(π_π)*r (k)
and π_π(π‘) = (πΆ_π)exp(βπΌπ‘(π_π)^2) (l)
Application to boundary and initial conditions. Conditions (c-1) and (c-2) give π΅_π = 0 and Biπ½0(π_π)π0 = (π_π)π0π½1*(π_π)*π0 (m) Where Bi is the Biot number defined as Bi = hr0 / k. The roots of (m) give the constants Ξ»k. Substituting (k) and (l) into (f) and summing all solutions
Application of the non-homogeneous initial condition (c-3) yields
The characteristic functions J0 ((Ξ»_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((Ξ»_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