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Mat-F February 9, 2005 Partial Differential Equations

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)

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Mat-F February 9, 2005 Partial Differential Equations

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  1. Mat-FFebruary 9, 2005Partial Differential Equations Åke Nordlund Niels Obers, Sigfus Johnsen Kristoffer Hauskov Andersen Peter Browne Rønne

  2. 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

  3. 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

  4. 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 F1(u) u/x + F2(u) u/y … = 0 F(u/x, 2u/x2, …) = 0

  5. 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, …) = 0

  6. Chapter 18 in Riley et al. • General and particular solutions • boundary conditions  particular solutions • Discussion of existence and uniqueness • characteristics • next week

  7. PDEs in Physics • Most common independent variables: • space and time {x,y,z,t}

  8. 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 … = 0 F(u/x, 2u/x2, …) = 0

  9. Important PDEs in Physics • Wave Equations • sound waves, light, matter waves, … 2u/t2 = c2 2u/x2

  10. Important PDEs in Physics • Wave Equations • sound waves, light, matter waves, … • Diffusion Equations • heat, viscous stress, magnetic diffusion, … u/t= 2u/x2

  11. 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 = 0 2u/x2 + 2u/y2 + 2u/z2 = 4πGρ

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

  13. Finding solutions from known PDEs • Harder! • Analytically • Manually, from rules, experience, known cases, ... • Computer programs (Maple, Mathematica, …)

  14. 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, …)

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

  16. 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

  17. Today • Finding PDEs from known solutions • explained here

  18. Today • Finding PDEs from known solutions • explained here • Test if expressions are solutions • straightforward

  19. 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)

  20. 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)

  21. Finding PDEs from known solutions • All three may be written as functions ofp(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)

  22. Finding solutions to PDEs • Wave equation 2u/t2 = c2 2u/x2 • Function of linear combination of x and t u = u1(x – c t) + u2(x + c t)

  23. Finding solutions to PDEs • Diffusion equation u/t= 2u/x2 • Need t-derivative same as 2nd space deriv.. u= e - a t sin(b x + c)

  24. Finding solutions to PDEs • First order PDEs Example: x u/x+ 3u = x2 Divide with x: u/x+ 3u/x = x Recognize x3u(multiply through) : (x3u)/x= x4 or:u= x2 /5 + f(y)/x3 Integrate : x3u= x5 /5 + f(y)

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

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