MA 201: Partial Differential Equations Lecture - 1 · Deﬁnition (Solution of a PDE) A function φ(x1,...,xn) is a solution to (2) if φ and its partial derivatives ... • The Laplace’s

Introduction and Basic DefinitionsFirst-Order Partial Differential Equations

MA 201: Partial Differential Equations

Lecture - 1

MA201(2014):PDE


Let us recall the general form of the nth order ordinary differentialequation(ODE):

F (x , y(x), y ′(x), y ′′(x), · · · , y (n)(x)) = 0, (1)

where y ′(x) = dy

dx, y ′′(x) = d2y

dx2, · · · , y (n)(x) = dny

dxn.

Facts:

• In an ODE, there is only one independent variable x so that all thederivatives appearing in the equation are ordinary derivatives of theunknown function y(x).

• The order of an ODE is the order of the highest derivative thatoccurs in the equation.

• Equation (1) is linear if F is linear in y , y ′, y ′′, . . . , y (n), with thecoefficients depending on the independent variable x .

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Definition (Partial Diff. Equations)A partial differential equation (PDE) for a function u(x1, x2, . . . , xn)(n ≥ 2) is a relation of the form

F (x1, x2, . . . , xn, u, ux1 , ux2 , . . . , ux1x1 , ux1x2 , . . . , ) = 0, (2)

where F is a given function of the independent variables x1, x2, . . . , xn; ofthe unknown function u and of a finite number of its partial derivatives.

Definition (Solution of a PDE)A function φ(x1, . . . , xn) is a solution to (2) if φ and its partial derivativesappearing in (2) satisfy (2) identically for x1, . . . , xn in some regionΩ ⊂ Rn.

The order of an equation: The order of a PDE is the order of thehighest derivative appearing in the equation. If the highest derivative isof order m, then the equation is said to be order m.

ut − uxx = f (x , t) (second-order equation)

ut + uxxx + uxxxx = 0 (fourth-order equation)

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Examples.

•

∂u

∂t= k

∂2u

∂x2, k ∈ R+: 2nd order equation in two variables x and t.

•

(

∂u

∂x

)3

+∂u

∂t= 0: 1st order third degree equation in two variables

x and t.

• x∂u

∂x+ y

∂u

∂y+

∂u

∂t= 0: 1st order equation in three variables x , y

and t.

•

∂3u

∂x3+

∂3u

∂x2∂y−

∂u

∂t= f (x , y , t): 3rd order equation in three

variables x , y and t.

•

∂2u

∂x2+

∂2u

∂y2+

∂2u

∂z2= 0: 2nd order equation in three variables x , y

and z .

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Definition (Classification)A PDE of order m is said to be

• linear if F is linear in the unknown function u and its partialderivatives, with coefficients depending only on the independentvariables x1, x2, . . . , xn.

Ex. x∂u

∂x+ y

∂u

∂y= xu + y , y

∂u

∂x− x

∂u

∂y= xyu + x .

• quasi-linear if it is linear in the derivatives of order m withcoefficients that depend on x1, x2, . . . , xn and the derivatives of order< m.

Ex. (u2 − xy)∂u

∂t+ u

∂u

∂x= 0.

• semi-linear if it is quasi-linear and the coefficients of derivatives oforder m are functions of the independent variables only.

Ex. (x2 + y2)∂u

∂t+ 2xy

∂u

∂x= 0.

• nonlinear if it is not linear in the derivatives of order m, e.g.,xu2x + yu2y = k .

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Example (Some well-known PDEs)

• The Laplace’s equation in n dimensions:

∆u :=

n∑

i=1

∂2u

∂x2i= 0 (second-order, linear, homogeneous)

• The Poisson equation:

∆u = f (second-order, linear, nonhomogeneous)

• The heat equation:

∂u

∂t−k∆u = 0 (k = const. > 0) (second-order, linear, homogeneous)

• The wave equation:

∂2u

∂t2−c2∆u = 0 (c = const. > 0) (second-order, linear, homogeneous)

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• The Transport equation:

∂u

∂t+

∂u

∂x= 0 (first-order, linear, homogeneous)

• The Burger’s equation:

∂u

∂t+ u

∂u

∂x= 0 (first-order, quasilinear, homogeneous)

What type of equations we will be studying?

• Linear, quasi-linear, and nonlinear first-order PDEs involving twoindependent variables.

• Linear second-order PDEs in two independent variables.

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First-order PDEs: A first order PDE in two independent variables x , yand the dependent variable u can be written in the form

F (x , y , u,∂u

∂x,∂u

∂y) = 0. (3)

For convenience, set

p =∂u

∂x, q =

∂u

∂y.

Equation (3) then takes the form

F (x , y , u, p, q) = 0. (4)

First-order PDEs arise in many applications, such as

• Transport of material in a fluid flow.

• Propagation of wave-fronts in optics.

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Classification of first-order PDEs

• If (3) is of the form

a(x , y)∂u

∂x+ b(x , y)

∂u

∂y= c(x , y)u + d(x , y)

then it is a linear first-order PDE.

• If (3) has the form

a(x , y)∂u

∂x+ b(x , y)

∂u

∂y= c(x , y , u)

then it is a semilinear first-order PDE. It is linear in the leading (highest-order)terms ∂u

∂xand ∂u

∂y. However, it need not be linear in u.

• If (3) has the form

a(x , y , u)∂u

∂x+ b(x , y , u)

∂u

∂y= c(x , y , u)

then it a quasi-linear PDE. Here F is linear in ∂u∂x

and ∂u∂y

with the coefficients a,

b and c depending on x and y as well as u.

• If F is not linear in the derivatives ∂u∂x

and ∂u∂y

, then (3) is said to be nonlinear.

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Linear PDE $ Semi-linear PDE $ Quasi-linear PDE $ PDE

Example

• xux + yuy = u (linear)

• xux + yuy = u2 (semi-linear)

• ux + (x + y)uy = xy (linear)

• uux + uy = 0 (quasi-linear)

• xu2x + yu2y = 2 (nonlinear)

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How first-order PDEs occur?

ExampleThe equation

x2 + y2 + (u − c)2 = r2, (5)

where r and c are arbitrary constants, represents the set of all sphereswhose centers lie on the u-axis. Differentiating (5) with respect to x , weobtain

x + (u − c)∂u

∂x= 0. (6)

Differentiating (5) with respect to y to have

y + (u − c)∂u

∂y= 0. (7)

Eliminating the arbitrary constant c from (6) and (7), we obtain thefirst-order PDE

y∂u

∂x− x

∂u

∂y= 0. (8)

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How first-order PDEs occur?

• Two-parameter family of surfaces: Let

f (x , y , u, a, b) = 0 (9)

represent two parameters family of surfaces in R3, where a and b arearbitrary constants.Differentiating (9) with respect to x and y yieldsa relations

∂f

∂x+ p

∂f

∂u= 0, (10)

∂f

∂y+ q

∂f

∂u= 0. (11)

Eliminating a and b from (9), (10) and (11), we get a relation of theform

F (x , y , u, p, q) = 0, (12)

which is a PDE for the unknown function u of two independentvariables.

• The applications of conservation principles often yield a first-orderPDEs.

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Unknown function of known functions

ExampleConsider a surface described by an equation of the form

u = f (x2 + y2), (13)

where f is an arbitrary function of a known function g(x , y) = x2 + y2.All surfaces of revolution about u-axis are characterized by theseequations.Differentiating (13) with respect to x and y , it follows that

ux = 2xf ′(g); uy = 2yf ′(g),

where f ′(g) =df

dg. Eliminating f ′(g) from the above two equations, we

obtain a first-order PDEyux − xuy = 0.

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Unknown function of known functions

• Unknown function of a single known function

Letu = f (g), (14)

where f is an unknown function and g is a known function of twoindependent variable x and y .Differentiating (14) with respect to x and y yield the equations

ux = f ′(g)gx (15)

anduy = f ′(g)gy , (16)

respectively. Eliminating f ′(g) from (19) and (20), we obtain

gyux − gxuy = 0,

which is a first-order linear PDE for u.

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Unknown functions of two known functions

A generalization of u = f (g) is

F (u, v) = 0, (17)

where u and v are known functions of x , y and z , and F is any functionof u and v .Differentiating (17) w.r.t. x and y , respectively, we get

∂F

∂u

∂u

∂x+ p

∂u

∂z

+∂F

∂v

∂v

∂x+ p

∂v

∂z

= 0,

∂F

∂u

∂u

∂y+ q

∂u

∂z

+∂F

∂v

∂v

∂y+ q

∂v

∂z

= 0.

If we, now, eliminate ∂F/∂u and ∂F/∂v from these equations, we obtainthe equation

p∂(u, v)

∂(y , z)+ q

∂(u, v)

∂(z , x)=

∂(u, v)

∂(x , y), (18)

which is a linear first-order pde.

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• Unknown functions of two known functions

Let

u = f (x − ay) + g(x + ay), (19)

where a > 0 is a constant. With v(x , y) = x − ay andw(x , y) = x + ay , we write (19) as

u = f (v) + g(w). (20)

Differentiating (20) w. r. t. x and y yields

p = ux = f ′(x − ay) + g ′(x + ay),

q = uy = −af ′(x − ay) + ag ′(x + ay)

Eliminating f ′(v) and g ′(w), we get

qy = a2px .

In terms of u, the above PDE is the well-known wave equation

uyy = a2uxx .

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Example (Geometrical problem)Suppose u = u(x , y) be a surface with the following property: At anypoint P(x0, y0, u(x0, y0)) on the surface, the plane tangent to the surfacepasses through the origin. What differential equation the surface shouldsatisfy?

The equation of the tangent plane at (x0, y0, u(x0, y0)) is

ux(x0, y0)(

x − x0)

+ uy (x0, y0)(

y − y0)

−(

u − u(x0, y0))

= 0.

Since this plane passes through the origin (0, 0, 0), we have

−ux(x0, y0)x0 − uy (x0, y0)y0 + u(x0, y0) = 0. (21)

For equation (21) to hold for all (x0, y0) in the domain of u, u mustsatisfy

xux + yuy − u = 0,

which is a first-order PDE.

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Cauchy’s problem or IVP for first-order PDEs: If

(a) x0(µ), y0(µ) and z0(µ) are functions which, together with their firstderivatives, are continuous in the interval M defined byµ1 < µ < µ2; and

(b) F (x , y , z , p, q) is a continuous function of x , y , z , p and q in acertain region U of the xyzpq space,

then it is required to establish the existence of a function φ(x , y) withthe following properties:

1 φ(x , y) and its partial derivatives with respect to x and y arecontinuous functions of x and y in a region R of the xy space.

2 For all values of x and y lying in R , the point(

x , y , φ(x , y), φx (x , y), φy (x , y))

lies in U and

F(

x , y , φ(x , y), φx (x , y), φy (x , y))

= 0.

3 For all µ belonging to the interval M , the point(

x0(µ), y0(µ))

belongs to the region R , and

φ(

x0(µ), y0(µ))

= z0(µ).

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Cauchy’s problem or IVP for first-order PDEs:From the point of view of geometry: whether there exists a surfacez = φ(x , y) which passes through the curve Γ whose parametric equationis

x = x0(µ), y = y0(µ), z = z0(µ), µ ∈ M ,

and at every point of which the direction (p, q,−1) of the normal is suchthat

F (x , y , z , p, q) = 0.

This is one of the various forms of Cauchy’s problem.The curve Γ :

(

x0(µ), y0(µ), z0(µ))

, µ ∈ M , is called the initial curve ofthe problem and the equation

φ(

x0(µ), y0(µ))

= z0(µ).

is called the initial condition (or side condition) of the problem.

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One Existence Theorem:

TheoremIf g(y) and all its derivatives are continuous for |y − y0| < δ, if x0 is agiven number, and z0 = g(y0), q0 = g ′(y0), and if F (x , y , z , q) and allits partial derivatives are continuous in a region S defined by

|x − x0| < δ, |y − y0| < δ, |q − q0| < δ,

then there exists a unique function φ(x , y) such that(a) φ(x , y) and all its partial derivatives are continuous in a region R

defined by |x − x0| < δ1, |y − y0| < δ1;(b) For all (x , y) ∈ R , z = φ(x , y) is a solution of the equation

∂z

∂x= F

(

x , y , z ,∂z

∂y

)

;

(c) For all values of y in the interval |y − y0| < δ1, φ(x0, y) = g(y).

The above theorem is due to Sonia Kowalewski and is also known asCauchy-Kowalewski theorem.

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• Well-posed Problem (In the sense of Hadamard)

The Cauchy’s problem (PDE + side condition) is said to be well-posed ifit satisfies the following criteria:

1 The solution must exist.

2 The solution should be unique.

3 The solution should depend continuously on the initial and/orboundary data.

If one or more of the conditions above does not hold, we say that theproblem is ill-posed.

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Documents

MA 201: Partial Differential Equations Lecture - 1 · Deﬁnition (Solution of a PDE) A function φ(x1,...,xn) is a solution to (2) if φ and its partial derivatives ... • The Laplace’s