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

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

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.

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

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.

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.

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:

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

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 .

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

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

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

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

Now allow To get Thus

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.

Using the definition of LT We have

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

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

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

Time delay Let

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

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

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

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

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

Solution B.C:

Second shifting theorem Gives With and Our solution is

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

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

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

Solution diverges as Therefore; By

Physical analysis .At all times

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

Define If we let get Setting

Yields Apply LT with Which is first order DE

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

In terms of Laguerre polynomials and normalizing get

Finite solution requirement and Using