1. First Order Partial Differential Equations
Method of characteristics
Web Lecture WI
H.M. Schuttelaars
May 15, 2018
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
May 15, 2018

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
May 15, 2018

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
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5. First Order Linear Partial Differential Equations
Definition of a first order linear PDE:
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6. First Order Linear Partial Differential Equations
Definition of a first order linear PDE:
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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>
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8. First Order Linear Partial Differential Equations
Example
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9. First Order Linear Partial Differential Equations
Plot the direction field:
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10. First Order Linear Partial Differential Equations
Plot the direction field: t x
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11. First Order Linear Partial Differential Equations
Plot the direction field: t x
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12. First Order Linear Partial Differential Equations
Direction field:
Through every point, a curve exists that is tangent to <a,b> everywhere. t x
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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
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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.
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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:
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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
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17. First Order Linear Partial Differential Equations
SHORT INTERMEZZO
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18. First Order Linear Partial Differential Equations
SHORT INTERMEZZO
Parameterization of a line in 2 dimensions
Parameter representation of a circle
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19. First Order Linear Partial Differential Equations
Or in 3 dimensions
Parameter representation of a helix
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20. First Order Linear Partial Differential Equations
Or in 3 dimensions
NOW BACK TO THE CHARACTERISTIC BASE CURVES
Parameter representation of a helix
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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
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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
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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:
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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:
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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:
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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
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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:
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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).
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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
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30. First Order Linear Partial Differential Equations
The equations for the characteristic base were solved to get
the base curves in the example:
Solving
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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
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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!
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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).
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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 .
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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).
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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.
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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
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38. First Order Linear Partial Differential Equations
EXAMPLE 1
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39. First Order Linear Partial Differential Equations
Consider with
The corresponding PDE reads:
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40. First Order Linear Partial Differential Equations
Consider with
Parameterize this initial curve with parameter l:
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41. First Order Linear Partial Differential Equations
Consider with
Parameterize this initial curve with parameter l:
Solve the characteristic equations with these initial conditions.
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42. First Order Linear Partial Differential Equations
Consider with
The (parameterized) solution reads:
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43. First Order Linear Partial Differential Equations
Visualize the solution for various values of l:
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44. First Order Linear Partial Differential Equations
When all values of l and s are considered, we get the solution surface:
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45. First Order Linear Partial Differential Equations
EXAMPLE 2
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46. First Order Linear Partial Differential Equations
PDE:
Initial condition:
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47. First Order Linear Partial Differential Equations
PDE:
Initial condition:
Char eqns:
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48. First Order Linear Partial Differential Equations
PDE:
Initial condition:
Char eqns:
Parameterised initial condition:
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49. First Order Linear Partial Differential Equations
Char eqns:
Parameterized initial condition:
Parameterized solution:
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50. First Order Linear Partial Differential Equations
Char eqns:
Initial condition:
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51. First Order Linear Partial Differential Equations
Char curves:
Initial condition:
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52. First Order Linear Partial Differential Equations
This gives the solution
surface, parameterised
by s and l.
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53. First Order Linear Partial Differential Equations
This gives the solution
surface, parameterised
by s and l.
This is the solution of
the original PDE
With initial condition
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54. First Order Linear Partial Differential Equations
Recipe to solve the PDE
To solve the partial differential equation explicitly, u(x,t) must be given at a certain curve C . As a first step, parameterize this curve with parameter l.
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55. First Order Linear Partial Differential Equations
Write down the characteristic equations, note that x, t and u now depend on both the parameters s and l.
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56. First Order Linear Partial Differential Equations
Solve the characteristic equations, using the conditions on the curve C . Take s=0 at this curve. Hence x, t and u are obtained as functions of s and l.
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57. First Order Linear Partial Differential Equations
Solve the characteristic equations, using the conditions on the curve C . Take s=0 at this curve. Hence x, t and u are obtained as functions of s and l.
To get u in terms of x and t, at least in the neighbourhood of C , explicit expressions for s and l are needed:
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58. First Order Linear Partial Differential Equations
Solve the characteristic equations, using the conditions on the curve C . Take s=0 at this curve. Hence x, t and u are obtained as functions of s and l.
To get u in terms of x and t, at least in the neighbourhood of C , explicit expressions for s and l are needed:
Implicit Function Theorem:
possible in a neighborhood of C if
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59. First Order Linear Partial Differential Equations
Solve the characteristic equations, using the conditions on the curve C . Take s=0 at this curve. Hence x, t and u are obtained as functions of s and l.
To get u in terms of x and t, at least in the neighbourhood of C , explicit expressions for s and l are needed: (Implicit Function Theorem)
possible in a neighborhood of C if
otherwise: no solution of infinitely many solutions
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60. First Order Linear Partial Differential Equations
EXAMPLE 3 (using Maple)
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61. First Order Linear Partial Differential Equations
Consider the PDV
with initial condition
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62. First Order Linear Partial Differential Equations
Consider the PDV
with initial condition
Visualize the IC using
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63. First Order Linear Partial Differential Equations
Consider the PDV
with initial condition
Write down the characteristic equations
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64. First Order Linear Partial Differential Equations
Consider the PDV
with initial condition
Solve the characteristic equations using dsolve:
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65. First Order Linear Partial Differential Equations
Consider the PDV
with initial condition
Solve the characteristic equations using dsolve:
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66. First Order Linear Partial Differential Equations
Use the solution sol
to vizualize the solution surface:
Note that in this example we can easily find s and l in terms of x and t and hence get the solution in terms of x and t:
s = t
l = x-t
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67. First Order Linear Partial Differential Equations
Use the solution sol
to plot the base characteristics:
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68. First Order Linear Partial Differential Equations
Use the solution sol
to plot the base characteristics:
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69. First Order Linear Partial Differential Equations
Use the solution sol
to vizualize the solution surface:
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70. First Order Linear Partial Differential Equations
Use the solution sol
to vizualize the solution surface:
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71. First Order Linear Partial Differential Equations
Use the solution sol
to vizualize the solution surface:
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72. First Order Linear Partial Differential Equations
Use the solution sol
to animate the solution in time:
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73. First Order Linear Partial Differential Equations
Use the solution sol
to animate the solution in time:
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74. First Order Linear Partial Differential Equations
EXAMPLE 4 (using Maple)
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75. First Order Linear Partial Differential Equations
Consider the PDV
with initial condition
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76. First Order Linear Partial Differential Equations
Consider the PDV
with initial condition
Write down the characteristic equations
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77. First Order Linear Partial Differential Equations
Using the same commands, we can find:
solution surface
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78. First Order Linear Partial Differential Equations
Using the same commands, we can find:
animation of the solution
solution surface
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79. Conclusions You are able to:
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
May 15, 2018

80. Conclusions You are able to:
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
Next lecture:
quasi-linear first order partial differential equations
May 15, 2018