Mat-F February 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, …) = 0 F 1 (u)  u/  x + F 2 (u)  u/  y … = 0 F(  u/  x,  2 u/  x 2, …) = 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  x u inst. of  u/  x can be easily generated in web pages (jfr. Mat-F netsted) F(  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) F 1 (u)  u/  x + F 2 (u)  u/  y … = 0 F(  u/  x,  2 u/  x 2, …) = 0

Important PDEs in Physics Wave Equations sound waves, light, matter waves, …  2 u/  t 2 = c 2  2 u/  x 2

Important PDEs in Physics Wave Equations sound waves, light, matter waves, … Diffusion Equations heat, viscous stress, magnetic diffusion, …  u/  t =   2 u/  x 2

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, …  2 u/  x 2 +  2 u/  y 2 +  2 u/  z 2 = 0  2 u/  x 2 +  2 u/  y 2 +  2 u/  z 2 = 4πGρ

…  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 F 1 (u)  u/  x + F 2 (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: u 1 (x,y) = x 4 + 4(x 2 y + y 2 + 1) u 2 (x,y) = sin(x 2 ) cos(2y) + cos(x 2 ) sin(2y) u 3 (x,y) = (x 2 +2y+2)/(3x 2 +6y+5) Examples: u 1 (x,y) = x 4 + 4(x 2 y + y 2 + 1) u 2 (x,y) = sin(x 2 ) cos(2y) + cos(x 2 ) sin(2y) u 3 (x,y) = (x 2 +2y+2)/(3x 2 +6y+5)

Finding PDEs from known solutions All three may be written as functions of p(x,y) = x 2 +2y Examples: u 1 (x,y) = x 4 + 4(x 2 y + y 2 + 1) = p u 2 (x,y) = sin(x 2 ) cos(2y) + cos(x 2 ) sin(2y) = sin(p) u 3 (x,y) = (x 2 +2y+2)/(3x 2 +6y+5) = (p+2)/(3p+5) Examples: u 1 (x,y) = x 4 + 4(x 2 y + y 2 + 1) = p u 2 (x,y) = sin(x 2 ) cos(2y) + cos(x 2 ) sin(2y) = sin(p) u 3 (x,y) = (x 2 +2y+2)/(3x 2 +6y+5) = (p+2)/(3p+5)

Finding solutions to PDEs Wave equation  2 u/  t 2 = c 2  2 u/  x 2 Wave equation  2 u/  t 2 = c 2  2 u/  x 2 Function of linear combination of x and t u = u 1 (x – c t) + u 2 (x + c t) Function of linear combination of x and t u = u 1 (x – c t) + u 2 (x + c t)

Finding solutions to PDEs Laplace equation  2 u/  x 2 +  2 u/  y 2 +  2 u/  z 2 = 0 Laplace equation  2 u/  x 2 +  2 u/  y 2 +  2 u/  z 2 = 0

Finding solutions to PDEs Diffusion equation  u/  t =   2 u/  x 2 Diffusion equation  u/  t =   2 u/  x 2 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 = x 2 Example: x  u/  x + 3u = x 2 Integrate : x 3 u = x 5 /5 + f(y) Integrate : x 3 u = x 5 /5 + f(y) Divide with x:  u/  x + 3u/x = x Divide with x:  u/  x + 3u/x = x Recognize x 3 u (multiply through) :  (x 3 u)/  x = x 4 Recognize x 3 u (multiply through) :  (x 3 u)/  x = x 4 or: u = x 2 /5 + f(y)/x 3

OK, we stop here! Good luck with the exercises 10:15-12:00