Partial Differential Equations-Differential Equations-Lecture Slides, Slides of Differential Equations and Transforms

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Mat-F

February 9, 2005

Partial Differential Equations

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

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

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, …  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

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)  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  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

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)