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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,
∂u
∂y= uy = u2, · · ·
(2)∂2u
∂x∂y= uyx = u21,
∂2u
∂x2= uxx = u11, · · ·
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,
∂u
∂y= uy = u2, · · ·
(2)∂2u
∂x∂y= uyx = u21,
∂2u
∂x2= uxx = u11, · · ·
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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
u∂u
∂x+ v
∂v
∂y= x − y ,
u∂u
∂y+ v
∂x
∂y= x + y ,
is a system of partial differential equations for the unknownfunctions u(x , y) and v(x , y).
Ayman Hashem Sakka Methods of Applied Mathematics
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
u∂u
∂x+ v
∂v
∂y= x − y ,
u∂u
∂y+ v
∂x
∂y= x + y ,
is a system of partial differential equations for the unknownfunctions u(x , y) and v(x , y).
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
Examples
Example
xux + yuy = ex is a PDE of order one.
Example
ut = uxxx + uxt is a third order PDE.
Ayman Hashem Sakka Methods of Applied Mathematics
Examples
Example
xux + yuy = ex is a PDE of order one.
Example
ut = uxxx + uxt is a third order PDE.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
Example
uuxx + uy = 0 is a quasilinear PDE of order two.
Example
utuxx + sin u ux = 0 is is not a quasilinear PDE.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
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.
Ayman Hashem Sakka Methods of Applied Mathematics
1.1 Heat Conduction
Ayman Hashem Sakka Methods of Applied Mathematics
1.1 Heat Conduction
Ayman Hashem Sakka Methods of Applied Mathematics