1. Int’l Workshop on SDE TT
COMSTECH, Islamabad, Pakistan
Jan. 30 – Feb. 1, 2014
Solution of Some Physical PDE by LT
Jamal Salah
Assistant Professor of Mathematics
A’Sharqiya University
Ibra, Oman

2. But he failed to invert transformation
DE started in 1675 by Newton and Bernoulli
J. Petzval (Geometrical optics) accredited LT!!
But he failed to invert transformation
Euler and Lagrange in 1744 considered similar transformation
Laplace in 1785 succeeded to invert it → LT

3. How to construct the DE? Why do we need DE's?
To describe a physical quantity of a system.
How it evolves with time.
How it varies with position.
How to construct the DE?
By using physical concepts, laws, or assumptions.
Conservation laws play a role in the construction.

4. Behavior of a physical quantity
Apply Concepts & Laws
System
Get DE
Solve it to get
Behavior of a physical quantity

5. Solving PDEs by Laplace Transform (LT) is
Powerful technique.
Converts DE to algebraic equation.
Can handle discontinuous functions.
Can be applied where separation of variables is ineffective.

6. Types of PDEs in Mathematical Physics are
Elliptic – Potential – FT
Parabolic – Heat diffusion – LT & FT
Hyperbolic – Wave Eqn. – LT & FT
They are covered by field equations with Initial and boundary conditions.

7. Physical Background Linear Time-Invariant theory (LTI theory)
LTI comes in applied mathematics with applications in NMR spectroscopy, seismology, circuits, signal processing & control theory … etc.
Linearity of LTI:

8. Time Invariance
Output does not depend on the particular time the input is applied
If input at t is has output
then at T seconds later
For LTI systems: output is the convolution

9. is response when transfer function F(s) H(s) Y(s)=H(s) F(s)
f(t) h(t) y(t)=f(t) *h(t)
impulse response .

10. Motivation of LT is for Linear Time-invariant Systems
Excitation Response
convolution
Thus
Eigen function Laplace Transform

11. Def. of LT S is a complex number
Connection of LT to Fourier transform FT:
LT is generalization to FT:
FT :

12. Thus LT is FT of modified version of
This means that (2)
When LT =FT
Convergence: Because of Eq. (2);
LT converges faster than FT
LT may converge when FT does not

13. Relation of LT to z-transform
For discrete-time input systems, we need z-transform defined as
For continuous-time systems: let

14. Now allow
To get Thus

15. By Def. LT is the expectation value of an exponential
Thus Relation to moments: Moments of a function are defined as: We will show that moments of a function are the expansion coefficients of its LT.

16. Using the definition of LT
We have

17. Therefore
Our result is Example from Electrostatics: A uniform charged rod of length L and charge Q on x-axis between a and b:

18. Charge in terms of step function
Model: linear charge density is Using LT Expanding

19. The zero moment:
This gives which is the just Q 1st moment: Electric dipole moment

20. Time delay
Let

21. LT advantage over S of V
Consider the DE With IC : and Separation of variables gives This solution does not satisfy the IC Using LT: Apply L to Eq. 3 and use

22. We get
Which gives Whose solution is Applying L to IC Thus:

23. Step function resolves earlier difficulties with boundary conditions
Inverse LT:
Step function
Step function resolves earlier difficulties with boundary conditions

24. Waggling semi-infinite string
We consider elastic string initially at rest and whose left end is moved sinusoidally. and

25. Wave equation
Initial conditions and Take LT w/r to t of wave equation 2st & 3nd terms are zeros whose solution is

26. Solution
B.C:

27. Second shifting theorem
Gives With and Our solution is

28. That is
Our solution demonstrates that no effect can reach the pint x in a time less than x/c. We may call the quantity

29. Heat equation by LT
A semi-infinite bar held at constant temperature . Then the end x=0 is held at zero temperature. Heat absorption = Heat flux or flow

30. Initial and boundary conditions
Are given by Apply LT to DE This is Inhomogeneous DE

31. Solution
diverges as Therefore; By

32. Physical analysis
.At all times

33. Harmonic Oscillator Potential in N Dimensions Via Laplace Transform
LT is used to reduce the N-dimensional Schrödinger equation into first order DE. The radial part of S. Eq. is: Harmonic Potential

34. Define
If we let get Setting

35. Yields
Apply LT with Which is first order DE

36. Letting -
Solution is Using inverse Laplace transform, the radial solution can be written as

37. In terms of Laguerre polynomials and normalizing
get

38. Finite solution requirement and
Using