MTH646 Official Handouts

MTH646 – PARTIAL DIFFERENTIAL EQUATIONS

Introduction:

In Mathematics, a partial differential equation is one of the types of differential equations, in which the equation contains unknown multi variables with their partial derivatives. It is a special case of an ordinary differential equation. Many world related problems in applied sciences, physics and engineering are modeled mathematically with partial differential equations. In numerical simulation, it is important to find out the exact solutions of partial differential equations but unfortunately we do not have appropriate methods to find the exact analytical solution of many types of partial differential equations. In this situation we utilize different approximations and other techniques to solve the problem numerically. There are several numerical methods are available in literature that help us to understand the mechanism and complexity of the differential problems.

Definition Partial Derivative:

6𝑢

u  Lim u(x  x, y, z, t)  u(x, y, z, t) x x0 x Provided that above limit exists.

u  Lim u(x, y  y, z, t)  u(x, y, z, t) y y0 y

u  Lim u(x, y, z  z, t)  u(x, y, z, t) Example: z z0 z Suppose 𝑢 is a function of more than one variable such that, u(x, y, z)  x cos z  x 2 y2e z u  cos z  2xy2 e z x u  0  x 22 ye z  2 yx2e z y u  x(sinz)  x 2 y2e z z

Partial Differential Equations:

A partial differential equation (PDE) is a relationship between an unknown function of several variables and its partial derivatives. Let 𝑢(𝑥, 𝑡) is unknown function and x and t are independent variables, then we

Examples: Heat Equation

Wave Equation

Laplace Equation

PDE involves two or more independent variables. In this example, 𝑥 and 𝑡 are independent variables.

Order of PDE:

The order of a PDE is the order of the highest derivative that occurs in the equation. u Examples: u u 1st order 1) t  x  0, order  order PDE nd 2u u 1/ 2 2 order 2)  (1 x2 y order  order PDE

4th order

Degree of a PDE:

4 u 3) x 2u t u 0, order  order PDE

A PDE is the degree of the highest order derivative which occurs in it after the equation has been rationalized. Examples: 1) u  t 2u c( x2 ),

Squaring, 2u 3 2) ( x2 )

3)

Squaring, u  ( u )  u  0, x 3u y x3 ,

3u 1 ( )2  y2 ( )2 ,

Dimension of PDE:

x x3

The dimension of a PDE is the number of independent variables taken in space direction in the partial differential equation. Examples: 1) u t 2 c dim  x2 2u 2u 2) x2  y2  0, dim  2u 2u 2u 3) x2 y2 z2  0, dim 

Linear and non-linear partial differential equation:

If the dependent variable and all its partial derivatives occur linearly in any PDE then such an equation is called linear PDE otherwise a nonlinear PDE. Examples:

1. (x 2  y2) u t 2u xy 3u 0
2. ux 2u x2  u 2xy2u xy  uy2u y2  (u )

x  (u )

y  u 3  0

2u

3. x2 2u 2 ( xy ) 2u y2 x 2  y2

Eq. (1) is linear but Eqs. (2) and (3) are non-linear partial differential equations.

Homogeneous and non-homogeneous PDEs:

If all the terms of a PDE contain the dependent variable or its partial derivatives then such a PDE is called homogeneous or non-homogeneous otherwise. Examples:

1) (x 2  y2) u t 2u xy 3u 0

2) ux 2u x2  u 2xy2u xy  uy2u y2  (u ) x  (u ) y  u 3  0

2u 3) x2 2u 2 ( xy ) 2u y2 x 2  y2

Eqs. (1) and (2) are homogeneous while Eq.(3) is non-homogeneous partial differential equations.

General Solutions of PDE:

A solution of a PDE is any function which satisfies the equation. A general solution of a PDE is a solution which contains the number of arbitrary independent functions equal to the order of the equation. 1 ( u )2 x Example: The solution of the second order PDE 2u xx  2x  y is

u  x2 y  1 xy2  F (x)  G( y)

Particular Solution of PDE:

A particular solution is one which can be obtained from general solution by particular choice of arbitrary function. Example:

Auxiliary Conditions:

In particular, F (x)  2 sin x, G(x)  3y4  5, then we have

u  x2 y  1 xy2  F (x)  G(x)

u  x2 y  1 xy2  2 sin x  3y4  5

Which is particular solution of the PDE.

The PDEs that represent physical systems usually have infinite number of solutions. For example: The functions u  x 2  y 2 , u  e x cos y, u  log(x 2  y 2 ), u  sin x.sinh y,... are entirely different from each other are solutions of PDE 2u   2u 

x y To obtain a unique (i.e. single) solution of the PDE corresponding to a given physical problem, one must use the additional information (i.e. auxiliary condition) arising from the physical situation. They fall in two categories:

1. Boundary conditions
2. Initial conditions

Boundary Conditions:

These are the conditions that must be satisfied at the points on the boundary S of the region R in which the partial differential equation hold. (or a condition that is required to be satisfied at all or part of the boundary S of a region R in which a set of differential conditions is to be solved).These are three types of boundary condition; ⚫ Dirichlet condition: u(x, y)  g(x, y) on S

⚫ Neumann ( or flax) condition:

6𝑢 u(x, y)  g(x, y) on S

n

Where 6𝑛 normal derivative i.e. a directional derivative taken in the direction normal to some surface.

⚫ Cauchy (or mixed) condition:

Initial Conditions:

(x, y)u(x, y)  (x, y) u(x, y)  g(x, y) on S

n

These are the conditions that must be satisfied throughout the region R at the instant when consideration of the physical system begins. When time t is one of the independent variable and we specify a condition at 𝑡 = 0, we refer to it an initial condition.

A typical initial condition is said to be of Cauchy type if the values of both u and

𝑡 = 0 i.e. the initial values of u and 6𝑢 are given. 6𝑡 The physical problems associated with PDE may be classified as:

•  Boundary-valued problems (B.V.P)
•  Initial-valued problems (I.V.P)
•  Initial-Boundary-valued problems (I.B.V.P) 6𝑢 6𝑡 on the boundary at