methods of applied mathematics -...

Methods of Applied Mathematics

Ayman Hashem Sakka

Islamic University of GazaFaculty of Science

Department of Mathematics

First Semester 2014-2015

Ayman Hashem Sakka Methods of Applied Mathematics

Introduction

Partial differential equations appear frequently in all areas ofphysics and engineering.

Notations

If u = u(x , y , · · · ), then we will use the following notations:

(1)∂u

∂x= ux = u1,

∂y= uy = u2, · · ·

(2)∂2u

∂x∂y= uyx = u21,

∂x2= uxx = u11, · · ·

Introduction

Partial differential equations appear frequently in all areas ofphysics and engineering.

Notations

If u = u(x , y , · · · ), then we will use the following notations:

(1)∂u

∂x= ux = u1,

∂y= uy = u2, · · ·

(2)∂2u

∂x∂y= uyx = u21,

∂x2= uxx = u11, · · ·

Partial differential equation

Definition

A partial differential equation (PDE) is an equation that containspartial derivatives of the unknown function.

The general form of a partial differential equation intwo independent variables x , y , and one dependentvariable u is

F (x , y , u, ux , uy , uxx , uxy , uyy , · · · ) = 0.

Partial differential equation

Definition

A partial differential equation (PDE) is an equation that containspartial derivatives of the unknown function.

The general form of a partial differential equation intwo independent variables x , y , and one dependentvariable u is

F (x , y , u, ux , uy , uxx , uxy , uyy , · · · ) = 0.

Examples

Example 1

xux − uuy = cos(xy)

is a partial differential equation for the unknown function u(x , y).

Example 2

The set of two equations

∂x+ v

∂y= x − y ,

∂y+ v

∂y= x + y ,

is a system of partial differential equations for the unknownfunctions u(x , y) and v(x , y).

Examples

Example 1

xux − uuy = cos(xy)

is a partial differential equation for the unknown function u(x , y).

Example 2

The set of two equations

∂x+ v

∂y= x − y ,

∂y+ v

∂y= x + y ,

is a system of partial differential equations for the unknownfunctions u(x , y) and v(x , y).

Solution of a partial differential equation

Remark

The unknown functions always depend on more than one variable.

Definition

A solution of a partial differential equation

F (x , y , u, ux , uy , uxx , uxy , uyy , · · · ) = 0

is a function u with continuous partial derivatives of all orders thatappear in the equation and that satisfies the differential equation atevery point of its domain of definition.

Solution of a partial differential equation

Remark

The unknown functions always depend on more than one variable.

Definition

A solution of a partial differential equation

F (x , y , u, ux , uy , uxx , uxy , uyy , · · · ) = 0

is a function u with continuous partial derivatives of all orders thatappear in the equation and that satisfies the differential equation atevery point of its domain of definition.

Initial and boundary conditions

Partial differential equations have in general infinitely manysolutions. In order to obtain a unique solution one mustsupplement the equation with additional conditions. There aretwo type of conditions commonly associated with partialdifferential equations:

(1) Initial conditions which give information about the solution u(or its derivative) at a given time t0.

(2) Boundary conditions which give information about the behaviorof the solution u (or its derivative) at the boundary of thedomain under consideration.

The conditions associated to a partial differential equationdepend on the type of the partial differential equation underconsideration.

Initial and boundary value problems

Definition

A partial differential equation with initial conditions is called aninitial value problem and a partial differential equation withboundary conditions is called a boundary value problem.

Classification of partial differential equations

Classification of partial differential equations paly a central rolein studying partial differential equations.

There exist several classifications. One classification isaccording to the order of the equation and anotherclassifications is according to linearity. Other importantclassifications will be described in later chapters.

Definition

(Order of a partial differential equation)The order of a partial differential equation is the order of the highestderivative that appear in the equation.

Classification of partial differential equations

Classification of partial differential equations paly a central rolein studying partial differential equations.

There exist several classifications. One classification isaccording to the order of the equation and anotherclassifications is according to linearity. Other importantclassifications will be described in later chapters.

Definition

(Order of a partial differential equation)The order of a partial differential equation is the order of the highestderivative that appear in the equation.

Examples

Example

xux + yuy = ex is a PDE of order one.

Example

ut = uxxx + uxt is a third order PDE.

Examples

Example

xux + yuy = ex is a PDE of order one.

Example

ut = uxxx + uxt is a third order PDE.

Linear partial differential equations

In calculus, analytic geometry and related areas, a linearfunction is a polynomial of degree one or less, including thezero polynomial.

For a function f (x1, · · · , xk) of any finite number independentvariables, the general formula is

f (x1, · · · , xk) = a1x1 + · · · + akxk + b,

and the graph is a hyperplane of dimension k .

Definition

A partial differential equation

F (x , y , u, ux , uy , · · · ) = 0

is called linear if F is linear in u, ux , uy , · · · ; that is if the dependentvariable and all its derivatives have degree one and are notmultiplied together. Otherwise it is called nonlinear.

Linear partial differential equations

In calculus, analytic geometry and related areas, a linearfunction is a polynomial of degree one or less, including thezero polynomial.

For a function f (x1, · · · , xk) of any finite number independentvariables, the general formula is

f (x1, · · · , xk) = a1x1 + · · · + akxk + b,

and the graph is a hyperplane of dimension k .

Definition

A partial differential equation

F (x , y , u, ux , uy , · · · ) = 0

is called linear if F is linear in u, ux , uy , · · · ; that is if the dependentvariable and all its derivatives have degree one and are notmultiplied together. Otherwise it is called nonlinear.

Remark

A functions F (x , y , u, ux , uy , · · · ) is linear in u, ux , uy , · · · impliesthat F is a sum of terms, each of which involves only the dependentvariable u or one of its derivatives to the first degree. There is norestriction on how the terms involve the independent variables.

Example

(1) ut = uxx + 2ux + u is a linear PDE of order 2.

(2) ut = uxxx + e−t is a linear PDE of order 3.

(3) uxuy = u is a nonlinear PDE of order 1.

(4) xuz + u2xy = z is a nonlinear PDE of order 2.

Remark

Example

Remark

Example

Remark

Example

Remark

Example

Remark

Example

Almost linear equations

Definition

A partial differential equation is called almost linear if thehighest-order derivatives have degree one and their coefficients arefunctions of the independent variables only.

Example

xuxx + ex+yuyy + uux = 0 is an almost linear PDE of order two.

Example

uux = x is is not an almost linear PDE.

Definition

Example

Definition

Example

Quasilinear equations

Definition

A partial differential equation is called quasilinear if thehighest-order derivatives have degree one and their coefficients arefunctions of u and the independent variables only.

Remark

A partial differential equation F (x , y , u, ux , uy , · · · ) = 0 is calledquasilinear if F is linear in highest-order derivatives included in theequation and the coefficients are functions of u and x , y only.

Quasilinear equations

Definition

A partial differential equation is called quasilinear if thehighest-order derivatives have degree one and their coefficients arefunctions of u and the independent variables only.

Remark

A partial differential equation F (x , y , u, ux , uy , · · · ) = 0 is calledquasilinear if F is linear in highest-order derivatives included in theequation and the coefficients are functions of u and x , y only.

Example

uuxx + uy = 0 is a quasilinear PDE of order two.

Example

utuxx + sin u ux = 0 is is not a quasilinear PDE.

Methods od solving partial differential equations

There are several methods to solve partial differential equations.Some of these methods are

(1) change of coordinates,

(2) separation of variables,

(3) integral transforms,

(4) numerical methods,

(5) perturbation methods,

(6) calculus of variations methods,

(7) integral equations.

Chapter 1The Diffusion Equation

In this chapter we study the diffusion equation

ut − (uxx + uyy + uzz) = p(x , y , z , t),

which describes a number of physical models, such as theconduction of heat in the a solid or the spread of acontaminant in a stationary medium.

We shall use this equation to introduce many of the solutiontechniques that will be useful in subsequent chapters in ourstudy of other types of linear partial differential equations.

To begin with, it is important to have a physical understandingof how the diffusion equation arises in a particular application,and we consider the simple model of heat conduction in a solid.

1.1 Heat Conduction

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