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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