First Order Partial Differential Equations

1. First Order Partial Differential Equations Method of characteristics Web Lecture WI2607-2008 H.M. Schuttelaars Delft Institute of Applied Mathematics

2. Contents • Linear First Order Partial Differential Equations • Derivation of the Characteristic Equation • Examples (solved using Maple) • Quasi-Linear Partial Differential Equations • Nonlinear Partial Differential Equations • Derivation of Characteristic Equations • Example

3. Contents • Linear First Order Partial Differential Equations • Derivation of the Characteristic Equation • Examples (solved using Maple) • Quasi-Linear Partial Differential Equations • Nonlinear Partial Differential Equations • Derivation of Characteristic Equations • Example

4. Contents • Linear First Order Partial Differential Equations • Derivation of the Characteristic Equation • Examples (solved using Maple) After this lecture: • you can recognize a linear first order PDE • you can write down the corresponding characteristic equations • you can parameterize the initial condition and solve the characteristic equation using the initial condition, either analytically or using Maple

5. First Order Linear Partial Differential Equations Definition of a first order linear PDE:

6. First Order Linear Partial Differential Equations Definition of a first order linear PDE:

7. First Order Linear Partial Differential Equations Definition of a first order linear PDE: This is the directional derivative of u in the direction <a,b>

8. First Order Linear Partial Differential Equations Example

9. First Order Linear Partial Differential Equations Plot the direction field:

10. First Order Linear Partial Differential Equations Plot the direction field: t x

11. First Order Linear Partial Differential Equations Plot the direction field: t x

12. First Order Linear Partial Differential Equations Direction field: Through every point, a curve exists that is tangent to <a,b> everywhere. t x

13. First Order Linear Partial Differential Equations Direction field: Through every point, a curve exists that is tangent to <a,b> everywhere: 1) Take points (0.5,0.5), (-0.1,0.5) and (0.2,0.01) X X X

14. First Order Linear Partial Differential Equations Direction field: Through every point, a curve exists that is tangent to <a,b>everywhere: 1) Take points (0.5,0.5), (-0.1,0.5) and (0.2,0.01) 2) Now draw the lines through those points that are tangent to <a,b> for all points on the lines.

15. First Order Linear Partial Differential Equations Zooming in on the line through (0.5,0.5), tangent to <a,b> for all x en t on the line: Direction field:

16. First Order Linear Partial Differential Equations Zooming in on the line through (0.5,0.5), tangent to <a,b> for all x en t on the line: Direction field: Parameterize these lines with a parameter s

17. First Order Linear Partial Differential Equations SHORT INTERMEZZO

18. First Order Linear Partial Differential Equations SHORT INTERMEZZO Parameterization of a line in 2 dimensions Parameter representation of a circle

19. First Order Linear Partial Differential Equations Or in 3 dimensions Parameter representation of a helix

20. First Order Linear Partial Differential Equations Or in 3 dimensions NOW BACK TO THE CHARACTERISTIC BASE CURVES Parameter representation of a helix

21. First Order Linear Partial Differential Equations • Parameterize these lines with a parameter s: Direction field: s=0.1 s=0.02 s=0 • For example: • s=0: • (x(0),t(0)) = (0.5,0.5) • changing s results in other points on this curve s=0.04 s=0.01

22. First Order Linear Partial Differential Equations • Parameterize these lines with a parameter s: • Its tangent vector is: Direction field: s=0.1 s=0.02 s=0 s=0.04 s=0.01

23. First Order Linear Partial Differential Equations • Parameterize these lines with a parameter s, x=x(s), t=t(s). • Its tangent vector is given by • On the curve:

24. First Order Linear Partial Differential Equations • Parameterize these lines with a parameter s, x=x(s), t=t(s). • Its tangent vector is given by • On the curve:

25. First Order Linear Partial Differential Equations • Parameterize these lines with a parameter s, x=x(s), t=t(s). • Its tangent vector is given by • On the curve:

26. First Order Linear Partial Differential Equations • Its tangent vector is given by • On the curve: IN WORDS: THE PDE REDUCES TO AN ODE ON THE CHARACTERISTIC CURVES

27. First Order Linear Partial Differential Equations • The PDE reduces to an ODE on the characteristic curves. • The characteristic equations (that define the characteristic curves) read:

28. First Order Linear Partial Differential Equations • The PDE reduces to an ODE on the characteristic curves. • The characteristic equations (that define the characteristic curves) read: One can solve for x(s) and t(s) without solving for u(s).

29. First Order Linear Partial Differential Equations • The PDE reduces to an ODE on the characteristic curves. • The characteristic equations (that define the characteristic curves) read: One can solve for x(s) and t(s) without solving for u(s). Gives the characteristic base curves

30. First Order Linear Partial Differential Equations The equations for the characteristic base were solved to get the base curves in the example: Solving

31. First Order Linear Partial Differential Equations The equations for the characteristic base were solved to get the base curves in the example: Solving gives

32. First Order Linear Partial Differential Equations This parameterisation, i.e., was plotted for (0.5,0.5) (x(0),t(0)) = (-0.1,0.5) (0.2,0.01) by varying s!

33. First Order Linear Partial Differential Equations To solve the original PDE, u(x,t) has to be prescribed at a certain curve C =C (x,t).

34. First Order Linear Partial Differential Equations To solve the original PDE, u(x,t) has to be prescribed at a certain curve C =C (x,t). The corresponding system of ODE’s has to be solved such that u(x,t) has the prescribed value at this curve C .

35. First Order Linear Partial Differential Equations The corresponding system of ODE’s has to be solved such that u(x,t) has the prescribed value at this curve C . • As a first step, parameterize the initial curve C with the parameter τ: x=x(τ), t=t(τ) and u=u(τ).

36. First Order Linear Partial Differential Equations The corresponding system of ODE’s has to be solved such that u(x,t) has the prescribed value at this curve C . • As a first step, parameterize the initial curve C with the parameter τ: x=x(τ), t=t(τ) and u=u(τ). • Next, the family of characteristic curves, determined by the points on C , may be parameterized by x=x(s, τ), t=t(x, τ) and u=u(s, τ), with the initial conditions prescribed for s=0.

37. First Order Linear Partial Differential Equations • As a first step, parameterize the initial curve C with the parameter τ: x=x(τ), t=t(τ) and u=u(τ). • Next, the family of characteristic curves, determined by the points on C , may be parameterized by x=x(s, τ), t=t(x, τ) and u=u(s, τ), with the initial conditions prescribed for s=0. This gives the solution surface

38. First Order Linear Partial Differential Equations EXAMPLE 1

39. First Order Linear Partial Differential Equations • Consider with The corresponding PDE reads:

40. First Order Linear Partial Differential Equations • Consider with • Parameterize this initial curve with parameter l:

41. First Order Linear Partial Differential Equations • Consider with • Parameterize this initial curve with parameter l: • Solve the characteristic equations with these initial conditions.

42. First Order Linear Partial Differential Equations • Consider with • The (parameterized) solution reads:

43. First Order Linear Partial Differential Equations Visualize the solution for various values of l:

44. First Order Linear Partial Differential Equations When all values of l and s are considered, we get the solution surface:

45. First Order Linear Partial Differential Equations EXAMPLE 2

46. First Order Linear Partial Differential Equations PDE: Initial condition:

47. First Order Linear Partial Differential Equations PDE: Initial condition: Char eqns:

48. First Order Linear Partial Differential Equations PDE: Initial condition: Char eqns: Parameterised initial condition:

49. First Order Linear Partial Differential Equations Char eqns: Parameterized initial condition: Parameterized solution:

50. First Order Linear Partial Differential Equations Char eqns: Initial condition: