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TERM PAPER of Complex analysis and transform techniques,MTH201

Topic: SHOW THAT LAPLACE EQUATION Uxx+Uyy=0 is elliptical

Submitted to: Submitted by:Mr. Rohit Gandhi Sir Manpreet singh

Roll.No. C6802B32 Reg.No-10805160

Class-C6802

Acknowledgement

I would like to thank the following people who have helped me in completing this project:-I would like to thank my teacher for giving me moral support and advising me wherever I was stuck. I would like to thank my parents, who gave me financial support and encouraged me in completing this project. I would also like to thank my friends whose support was always there for me.

INRTODUCTIONuxx + uyy = 0

Given a scalar field j, the Laplace equation in Cartesian coordinates is

 

A function which satisfies Laplace's equation is said to be harmonic. A solution to

Laplace's equation has the property that the average value over a spherical surface is

equal to the value at the center of the sphere (Gauss's harmonic function theorem).

Solutions have no local maxima or minima. Because Laplace's equation is linear, the

superposition of any two solutions is also a solution. Typically we are given a set of

boundary conditions and we need to solve for the (unique) scalar field j that is a

solution of the Laplace equation and that satisfies those boundary conditions.  dyA

solution of Laplace's equation is called a "harmonic functionThis is the two

dimensional Laplace.s equation after the french mathematician Pierre Laplace (1749-

1827. Solutions to this

equation are called harmonic functions. They play a role in problems

of heat conduction, .uid .ow, ...

Now we will prove that how LAPLACE equation Uxx + Uyy=0 is ELLIPTICAL...

Laplace Equation

second-order partial differential equation widely useful in physics because its

solutions R (known as harmonic functions) occur in problems of electrical, magnetic,

and gravitational potentials, of steady-state temperatures, and of hydrodynamics. The

equation was discovered by the French mathematician and astronomer Pierre-Simon

Laplace (1749–1827).

Laplace’s equation states that the sum of the second-order partial derivatives of R, the

unknown function, with respect to the Cartesian coordinates, equals zero:

The sum on the left often is represented by the expression ∇2R, in which the symbol ∇2 is called the Laplacian, or the Laplace operator.

three dimensions, the problem is to find twice differentiable real-valued functions ,

of real variables x, y, and z, such that

This is often written as

Or

where is the divergence, and is the gradient,or

where Δ is the laplace operator.

Boundary conditions

The Dirichlet problem for Laplace's equation consists of finding a solution on some

domain D such that on the boundary of D is equal to some given function. Since

the Laplace operator appears in the heat equation, one physical interpretation of this

problem is as follows: fix the temperature on the boundary of the domain according to

the given specification of the boundary condition. Allow heat to flow until a

stationary state is reached in which the temperature at each point on the domain

doesn't change anymore. The temperature distribution in the interior will then be

given by the solution to the corresponding Dirichlet problem.

The Neumann boundary conditions for Laplace's equation specify not the function

itself on the boundary of D, but its normal derivative. Physically, this corresponds to

the construction of a potential for a vector field whose effect is known at the boundary

of D alone.

Solutions of Laplace's equation are called harmonic functions; they are all analytic

within the domain where the equation is satisfied. If any two functions are solutions to

Laplace's equation (or any linear homogenous differential equation), their sum (or any

linear combination) is also a solution. This property, called the principle of

superposition, is very useful, e.g., solutions to complex problems can be constructed

by summing simple solutions.

ProveThe partial differential equations we will study are all linear equations of

second order; where the unknown function u is a function of two independent

variables x and y (or x and t if time t is involved, and there is only one space

dimension). This means they can be written in the form

Auxx + Buxy + Cuyy + Dux + Euy + Fu = G (1)

where A;B; ¢ ¢ ¢ ;G can be arbitrary functions of x and y. Any equation that

can be put in this form is called linear, and if G(x; y) ´ 0 it is called a

homogeneous linear p.d.e.. We will mainly be concerned with homogeneous

equations where the functions A;B; ¢ ¢ ¢ ; F are constants. These equations can

be classified into three different types, depending on the coefficients of the second-

order derivatives, as follows:

² If B2 - 4AC < 0, equation (1) is called an elliptic p.d.e.

² If B2 - 4AC = 0, equation (1) is called a parabolic p.d.e

.

² If B2-4AC > 0, equation (1) is called a hyperbolic p.d.e.

We will see that the methods used to solve the equation depend on which

type it is. We will study a simple equation of each type. The elliptic equation

we will solve is Laplace’s equation uxx + uyy = 0. The one-dimensional

heat equation ut = ®2uxx will be our example of a parabolic p.d.e., and the

wave equation utt = b2uxx will be the hyperbolic example.

Example 1 Show that u (x; y) = ex sin y is a solution of Laplace.s equation in

two dimensions.

For this, we compute uxx and uyy and show that uxx + uyy = 0.

ux = ex sin y

uxx = ex sin y

uy = ex cos y

uyy = -ex sin y

Therefore

uxx + uyy = ex sin y - ex sin y

= 0

Example 2 uxx + uyy = 0. In this case B2 - 4AC = 0 - 4 (1) (1) = -4.

Thus it is elliptic.

APPLICATIONS

Laplace equation in two dimensionsThe Laplace equation in two independent variables has the form

Analytic functionsThe real and imaginary parts of a complex analytic functions both satisfy the Laplace

equation. That is, if z = x + iy, and if

then the necessary condition that f(z) be analytic is that the C.R equations be satisfied:

where ux is the first partial derivative of u with respect to x.

It follows that

Therefore u satisfies the Laplace equation. A similar calculation shows that v also

satisfies the Laplace equation.

Conversely, given a harmonic function, it is the real part of an analytic function, f(z)

(at least locally). If a trial form is

then the Cauchy-Riemann equations will be satisfied if we set

The Laplace equation for φ implies that the integrability condition for ψ is satisfied:

and thus ψ may be defined by a line integral. The integrability condition and Stoke’s

therom implies that the value of the line integral connecting two points is independent

of the path. The resulting pair of solutions of the Laplace equation are called

conjugate harmonic functions. This construction is only valid locally, or provided that

the path does not loop around a singularity. For example, if r and θ are polar

coordinates and

then a corresponding analytic function is

However, the angle θ is single-valued only in a region that does not enclose the origin.

The close connection between the Laplace equation and analytic functions implies

that any solution of the Laplace equation has derivatives of all orders, and can be

expanded in a power series, at least inside a circle that does not enclose a singularity.

This is in sharp contrast to solutions of the Wave equations , which generally have

less regularity.

There is an intimate connection between power series and Fourier analysis . If we

expand a function f in a power series inside a circle of radius R, this means that

with suitably defined coefficients whose real and imaginary parts are given by

Therefore

which is a Fourier series for f.

Fluid flow Let the quantities u and v be the horizontal and vertical components of the velocity

field of a steady incompressible, irrotational flow in two dimensions. The condition

that the flow be incompressible is that

and the condition that the flow be irrotational is that

If we define the differential of a function ψ by

then the incompressibility condition is the integrability condition for this differential:

the resulting function is called the stream function because it is constant along lines

flow. The first derivatives of ψ are given by

and the irrotationality condition implies that ψ satisfies the Laplace equation. The

harmonic function φ that is conjugate to ψ is called the velocity potential. The

Cauchy-Riemann equations imply that

Thus every analytic function corresponds to a steady incompressible, irrotational fluid

flow in the plane. The real part is the velocity potential, and the imaginary part is the

stream function.

Bibliography. dip.sun.ac.za/~hanno/twb264/lesings/laplace_eqn.pdf

. dmpeli.mcmaster.ca/TeachProjects/Math3D03/.../CourseWare52.pdf

. www.stanford.edu/class/math220a/handouts/secondorder.pdf

. www.math.osu.edu/~hines/files/415/ laplace rect.pdf

Books

.N.P bali

.G,s garewall