Mat-FFebruary 9, 2005

Partial Differential Equations

Åke Nordlund

Niels Obers, Sigfus Johnsen

Kristoffer Hauskov Andersen

Peter Browne Rønne

Exercises Today:Maple T.A.

Register Name: exactly as under ISIS! Student ID: phone number

Quiz: Part I Multiple selection (1 of 2)

Anonymous (“flash card”) training Mastery: Part II

2 problems

Structure and Schedule(see also the SIS-web)

Monday Lecture + Exercise (2+2)

some turn-in-assignments (paper)

Wednesday Lecture (9-10?) + Exercise (1+2)

computer-aided (with Maple & Maple T.A.(?) ) problems = turn-in-assignments (Maple)

Self-studies repeat + material for next Monday

Partial Differential Equations(PDEs)

Relate an unknown function u(x,y,…) of two or more variable to its partial derivatives with respect to those variables; e.g.,

We will be looking mostly at linear PDEs 1st and 2nd order PDEs

F(u/x, u/y, …, u, …) = 0F(u/x, u/y, …, u, …) = 0

F1(u) u/x + F2(u) u/y … = 0F1(u) u/x + F2(u) u/y … = 0

F(u/x, 2u/x2, …) = 0F(u/x, 2u/x2, …) = 0

Partial Differential Equations(PDEs)

Relate an unknown function u(x,y,…) of two or more variable to its partial derivatives with respect to those variables; e.g.,

We often use the notation xu inst. of u/x can be easily generated in web pages

(jfr. Mat-F netsted)

F(u/x, u/y, …, u, …) = 0F(u/x, u/y, …, u, …) = 0

Chapter 18 in Riley et al.

General and particular solutions boundary conditions particular solutions

Discussion of existence and uniqueness characteristics

next week

PDEs in Physics

Most common independent variables: space and time {x,y,z,t}

PDEs in Physics

Most common independent variables: space and time {x,y,z,t}

Most common form of PDEs: linear (no squares of partial derivatives)

2nd order (up to 2nd derivatives w.r.t. indep. vars)F1(u) u/x + F2(u) u/y … = 0F1(u) u/x + F2(u) u/y … = 0

F(u/x, 2u/x2, …) = 0F(u/x, 2u/x2, …) = 0

Important PDEs in Physics

Wave Equations sound waves, light, matter waves, …

2u/t2 = c2 2u/x22u/t2 = c2 2u/x2

Important PDEs in Physics

Wave Equations sound waves, light, matter waves, …

Diffusion Equations heat, viscous stress, magnetic diffusion, …

u/t = 2u/x2u/t = 2u/x2

Important PDEs in Physics

Wave Equations sound waves, light, matter waves, …

Diffusion Equations heat, viscous stress, magnetic diffusion, …

Laplace and Poisson Equations gravity, electric potential, …

2u/x2 + 2u/y2 + 2u/z2 = 02u/x2 + 2u/y2 + 2u/z2 = 02u/x2 + 2u/y2 + 2u/z2 = 4πGρ2u/x2 + 2u/y2 + 2u/z2 = 4πGρ

… u/x + … u/y … = 0… u/x + … u/y … = 0

Finding a PDE from known solutions

Suppose you have u(x,y) and you want to know which PDE it might obey …

take partial derivatives see how you can combine & cancel them

F1(u) u/x + F2(u) u/y … = 0F1(u) u/x + F2(u) u/y … = 0

Finding solutions from known PDEs

Harder!

Analytically Manually, from rules, experience, known cases, ... Computer programs (Maple, Mathematica, …)

Finding solutions from known PDEs

Harder!

Analytically Manually, from rules, experience, known cases, ... Computer programs (Maple, Mathematica, …)

Numerically Tool programs (Maple, Mathematica, …) Programming languages + methods (Numerical

Recipes, …)

Exercises

Mondays; analytical work (manual mostly) groups are now assigned (was delayed by ISIS) it is OK to trade groups (use the ISIS mechanism)

Exercises

Mondays; analytical work (manual mostly) groups are now assigned (was delayed by ISIS) it is OK to trade groups (use the ISIS mechanism)

Wednesdays; computer-aided Maple Maple T.A. (if we can get it – was promised)

problem posing; individual variations interactive problem solving semi-automatic grading

Today

Finding PDEs from known solutions explained here

Today

Finding PDEs from known solutions explained here

Test if expressions are solutions straightforward

Today

Finding PDEs from known solutions explained here

Test if expressions are solutions straightforward

Find solutions to PDEs by combining partial derivatives (trial and error)

Finding PDEs from known solutions

Check if suggested solutions may be written as functions of a single p(x,y)

Examples:

u1(x,y) = x4 + 4(x2y + y2 + 1)

u2(x,y) = sin(x2) cos(2y) + cos(x2) sin(2y)

u3(x,y) = (x2+2y+2)/(3x2+6y+5)

Examples:

u1(x,y) = x4 + 4(x2y + y2 + 1)

u2(x,y) = sin(x2) cos(2y) + cos(x2) sin(2y)

u3(x,y) = (x2+2y+2)/(3x2+6y+5)

Finding PDEs from known solutions

All three may be written as functions of p(x,y) = x2+2y

Examples:

u1(x,y) = x4 + 4(x2y + y2 + 1)

= p2 + 4

u2(x,y) = sin(x2) cos(2y) + cos(x2) sin(2y)

= sin(p)

u3(x,y) = (x2+2y+2)/(3x2+6y+5)

= (p+2)/(3p+5)

Examples:

u1(x,y) = x4 + 4(x2y + y2 + 1)

= p2 + 4

u2(x,y) = sin(x2) cos(2y) + cos(x2) sin(2y)

= sin(p)

u3(x,y) = (x2+2y+2)/(3x2+6y+5)

= (p+2)/(3p+5)

Finding solutions to PDEs

Wave equation

2u/t2 = c2 2u/x2

Wave equation

2u/t2 = c2 2u/x2

Function of linear combination of x and t

u = u1(x – c t) + u2(x + c t)

Function of linear combination of x and t

u = u1(x – c t) + u2(x + c t)

Finding solutions to PDEs

Diffusion equation

u/t = 2u/x2

Diffusion equation

u/t = 2u/x2

Need t-derivative same as 2nd space deriv..

u = e - a t sin(b x + c)

Need t-derivative same as 2nd space deriv..

u = e - a t sin(b x + c)

Finding solutions to PDEs

First order PDEs

Example:

x u/x + 3u = x2

Example:

x u/x + 3u = x2

Integrate :

x3u = x5 /5 + f(y)

Integrate :

x3u = x5 /5 + f(y)

Divide with x:

u/x + 3u/x = x

Divide with x:

u/x + 3u/x = x

Recognize x3u (multiply through) :

(x3u)/x = x4

Recognize x3u (multiply through) :

(x3u)/x = x4

or: u = x2 /5 + f(y)/x3or: u = x2 /5 + f(y)/x3

OK, we stop here!

Good luck with the exercises 10:15-12:00Good luck with the exercises 10:15-12:00