1. Study of Denominated Linear SO-ODEs
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Models Highly Useful for Design of Fins….

2. Denominated Linear SO-ODEs
Second order linear equations with variable coefficients are in general more difficult to solve than equations with constant coefficients.
Great mathematicians invented solutions for SO-ODEs with specific functions as variable coefficients.
These are:
The Euler equidimensional Equation.
Sturm-Liouville Differential Equations.
Bessel Differential Equation.
Legendre’s Differential equation.
Chebyshev’s differential equation

3. Euler Equidimensional Equation
The Euler Equidimensional Equation is a special equation.
The ideas, used to solve second order linear equations with constant coefficients can be used to solve EEE.
This equation is also called Cauchy equidimensional equation, Cauchy equation, Cauchy-Euler equation.
Euler’s inventions were so extensive that many 20th Century mathematicians tried to avoid confusion by naming these inventions after the person who first studied them after Euler.

4. Euler's prolific output caused a tremendous problem of backlog.
Leonhard Euler
Leonhard Euler ( ) was arguably the greatest mathematician of the eighteenth century.
One of the most prolific of all time; his publication list of 886 papers and books fill about 90 volumes.
Remarkably, much of this output dates from the the last two decades of his life, when he was totally blind.
Euler's prolific output caused a tremendous problem of backlog.
The St. Petersburg Academy continued publishing his work posthumously for more than 30 years.

5. Euler Equidimensional Equation
The Euler equidimensional equation for the unknown y with singular point at x0  is given by the equation
where k1 and k0 are constants.

6. The concept of Equidimensional Nature
The equation is called equidimensional because if the variable x has any physical dimensions, then the terms with
For any nonnegative integer n, all terms are actually dimensionless.
The exponential functions y(x) = ex are not valid Ansaz to the EEE.
It is simple to show that there is no constant  such that the exponential is solution.
Euler introduced a new Ansaz.

7. Original Method of Solution
Consider the Euler Equidimensional Equation
where k1, k0 and x0 are real constants.
Euler thought an inverse of Method of characteristics.
Invented a quadratic equation for which  roots are influenced by k1 & k0 .

8. General Solution to Euler Equation
If + -, real or complex, then the general solution of Euler Equation is given by
If + = -, , then the general solution of Euler Equation is given by

9. The Sturm–Liouville Theory
Swiss mathematician Charles-François Sturm (1803–1855) and the French mathematician Joseph Liouville (1809–1882).
These mathematicians studied a certain second-order linear differential equations under appropriate boundary conditions and the properties of their solutions.
These are known as Sturm-Liouville differential equations or Sturm-Liouville Boundary Value Problem (SL-BVP).

10. Sturm-Liouville Differential Equation
A Sturm-Liouville equation is a second order linear differential equation that can be written in the form
Such an equation is said to be in Sturm-Liouville form.
When, p, q and are specified such that p(x) > 0 and (x)  0  x [a,b].
is known as spectral parameter.
It is frequently replaced by other variables or expressions.
where

11. SL-DE -1 : Bessel Differential Equation
A SL-DE is called as Bessel equation when
The parameter
 The Bessel equation is

12. Fins with Cylindrical Geometry Heat Transfer
Circumferential fin of rectangular profile

13. Zeroth Order Bessel Functions
m is called as order of the Bessel Function
For A Circumferential fin of rectangular profile

14. SL-DE -2 : Legendre’s differential equation
A SL-DE is called as Legendre’s differential equation when
The parameter
 The Legendre’s differential equation is

15. SL-DE -3 : Chebyshev’s differential equation
A SL-DE is called as Chebyshev’s differential equation when
The parameter
 The Chebyshev’s differential equation is

16. Second Generation Solution to EEE
Simplified form of EEE is developed by equating x0 = 0.
The general case x0  0 follows this case.
or, in standard form
Its hard to think of some likely candidates for particular solutions of above Equation.
A simple power law function is a natural ansatz for above ODE.
An educated guess for the solution is y = x, where  is a constant to be determined.