First Order Partial Differential Equations Method of characteristics

First Order Partial Differential Equations Method of characteristics Web Lecture WI 2607 -2008 H. M. Schuttelaars 12/3/2020 1 Delft Institute of Applied Mathematics

Contents • Linear First Order Partial Differential Equations Ø Derivation of the Characteristic Equation Ø Examples (solved using Maple) • Quasi-Linear Partial Differential Equations • Nonlinear Partial Differential Equations Ø Derivation of Characteristic Equations Ø Example 12/3/2020 2

Contents • Linear First Order Partial Differential Equations Ø Derivation of the Characteristic Equation Ø Examples (solved using Maple) • Quasi-Linear Partial Differential Equations • Nonlinear Partial Differential Equations Ø Derivation of Characteristic Equations Ø Example 12/3/2020 3

Contents • Linear First Order Partial Differential Equations Ø Derivation of the Characteristic Equation Ø 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 12/3/2020 4

First Order Linear Partial Differential Equations Definition of a first order linear PDE: 12/3/2020 5

First Order Linear Partial Differential Equations Definition of a first order linear PDE: 12/3/2020 6

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> 12/3/2020 7

First Order Linear Partial Differential Equations Example 12/3/2020 8

First Order Linear Partial Differential Equations Plot the direction field: 12/3/2020 9

First Order Linear Partial Differential Equations Plot the direction field: t x 12/3/2020 10

First Order Linear Partial Differential Equations Plot the direction field: t x 12/3/2020 11

First Order Linear Partial Differential Equations Direction field: Through every point, a curve exists that is tangent t to <a, b> everywhere. x 12/3/2020 12

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 12/3/2020 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) 2) Now draw the lines through those points that are tangent to <a, b> for all points on the lines. 12/3/2020 14

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: 12/3/2020 Direction field: 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: Parameterize these lines with a parameter s 12/3/2020 16

First Order Linear Partial Differential Equations SHORT INTERMEZZO 12/3/2020 17

First Order Linear Partial Differential Equations SHORT INTERMEZZO Parameterization of a line in 2 dimensions Parameter representation of a circle 12/3/2020 18

First Order Linear Partial Differential Equations Or in 3 dimensions Parameter representation of a helix 12/3/2020 19

First Order Linear Partial Differential Equations C I T S I R E T C Or in 3 dimensions A R A H S C E E H V T R O U T C K E S C A A B B W NO Parameter representation of a helix 12/3/2020 20

First Order Linear Partial Differential Equations • Parameterize these lines with a parameter s: Direction field: s=0. 1 s=0. 02 For example: • s=0: (x(0), t(0)) = (0. 5, 0. 5) • changing s results in other points on this curve 12/3/2020 s=0. 01 s=0. 04 21

First Order Linear Partial Differential Equations • Parameterize these lines with a parameter s: Direction field: s=0. 1 s=0. 02 • Its tangent vector is: s=0. 01 12/3/2020 s=0. 04 22

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: 12/3/2020 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: 12/3/2020 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: 12/3/2020 25

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 12/3/2020 26

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: 12/3/2020 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: One can solve for x(s) and t(s) without solving for u(s). 12/3/2020 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) Gives the characteristic and t(s) without solving base curves for u(s). 12/3/2020 29

First Order Linear Partial Differential Equations The equations for the characteristic base were solved to get the base curves in the example: Solving 12/3/2020 30

First Order Linear Partial Differential Equations The equations for the characteristic base were solved to get the base curves in the example: Solving 12/3/2020 gives 31

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! 12/3/2020 32

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). 12/3/2020 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). The corresponding system of ODE’s has to be solved such that u(x, t) has the prescribed value at this curve C. 12/3/2020 34

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). 12/3/2020 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). • 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. 12/3/2020 36

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 12/3/2020 37

First Order Linear Partial Differential Equations EXAMPLE 1 12/3/2020 38

First Order Linear Partial Differential Equations • Consider with The corresponding PDE reads: 12/3/2020 39

First Order Linear Partial Differential Equations • Consider with • Parameterize this initial curve with parameter l: 12/3/2020 40

First Order Linear Partial Differential Equations • Consider with • Parameterize this initial curve with parameter l: • Solve the characteristic equations with these initial conditions. 12/3/2020 41

First Order Linear Partial Differential Equations • Consider with • The (parameterized) solution reads: 12/3/2020 42

First Order Linear Partial Differential Equations Visualize the solution for various values of l: 12/3/2020 43

First Order Linear Partial Differential Equations When all values of l and s are considered, we get the solution surface: 12/3/2020 44

First Order Linear Partial Differential Equations EXAMPLE 2 12/3/2020 45

First Order Linear Partial Differential Equations PDE: Initial condition: 12/3/2020 46

First Order Linear Partial Differential Equations PDE: Initial condition: Char eqns: 12/3/2020 47

First Order Linear Partial Differential Equations PDE: Initial condition: Char eqns: Parameterised initial condition: 12/3/2020 48

First Order Linear Partial Differential Equations Char eqns: Parameterized initial condition: Parameterized solution: 12/3/2020 49

First Order Linear Partial Differential Equations Char eqns: Initial condition: 12/3/2020 50

First Order Linear Partial Differential Equations Char curves: Initial condition: 12/3/2020 51

First Order Linear Partial Differential Equations This gives the solution surface, parameterised by s and l. 12/3/2020 52

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 12/3/2020 53

First Order Linear Partial Differential Equations Recipe to solve the PDE 1. 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. 12/3/2020 54

First Order Linear Partial Differential Equations 2. Write down the characteristic equations, note that x, t and u now depend on both the parameters s and l. 12/3/2020 55

First Order Linear Partial Differential Equations 3. 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. 12/3/2020 56

First Order Linear Partial Differential Equations 3. 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. 4. To get u in terms of x and t, at least in the neighbourhood of C , explicit expressions for s and l are needed: 12/3/2020 57

First Order Linear Partial Differential Equations 3. 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. 4. 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 12/3/2020 58

First Order Linear Partial Differential Equations 3. 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. 4. 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 12/3/2020 59

First Order Linear Partial Differential Equations EXAMPLE 3 (using Maple) 12/3/2020 60

First Order Linear Partial Differential Equations Consider the PDV with initial condition 12/3/2020 61

First Order Linear Partial Differential Equations Consider the PDV with initial condition Visualize the IC using 12/3/2020 62

First Order Linear Partial Differential Equations Consider the PDV with initial condition Write down the characteristic equations 12/3/2020 63

First Order Linear Partial Differential Equations Consider the PDV with initial condition Solve the characteristic equations using dsolve: 12/3/2020 64

First Order Linear Partial Differential Equations Consider the PDV with initial condition Solve the characteristic equations using dsolve: 12/3/2020 65

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 12/3/2020 66

First Order Linear Partial Differential Equations Use the solution sol to plot the base characteristics: 12/3/2020 67

First Order Linear Partial Differential Equations Use the solution sol to plot the base characteristics: 12/3/2020 68

First Order Linear Partial Differential Equations Use the solution sol to vizualize the solution surface: 12/3/2020 69

First Order Linear Partial Differential Equations Use the solution sol to vizualize the solution surface: 12/3/2020 70

First Order Linear Partial Differential Equations Use the solution sol to vizualize the solution surface: 12/3/2020 71

First Order Linear Partial Differential Equations Use the solution sol to animate the solution in time: 12/3/2020 72

First Order Linear Partial Differential Equations Use the solution sol to animate the solution in time: 12/3/2020 73

First Order Linear Partial Differential Equations EXAMPLE 4 (using Maple) 12/3/2020 74

First Order Linear Partial Differential Equations Consider the PDV with initial condition 12/3/2020 75

First Order Linear Partial Differential Equations Consider the PDV with initial condition Write down the characteristic equations 12/3/2020 76

First Order Linear Partial Differential Equations Using the same commands, we can find: solution surface 12/3/2020 77

First Order Linear Partial Differential Equations Using the same commands, we can find: solution surface 12/3/2020 animation of the solution 78

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 12/3/2020 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 Next lecture: • quasi-linear first order partial differential equations 12/3/2020 80