Create Presentation
Download Presentation

Download Presentation
## First Order Partial Differential Equations

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**First Order Partial Differential Equations**Method of characteristics Web Lecture WI2607-2008 H.M. Schuttelaars Delft Institute of Applied Mathematics**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**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**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**First Order Linear Partial Differential Equations**Definition of a first order linear PDE:**First Order Linear Partial Differential Equations**Definition of a first order linear PDE:**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>**First Order Linear Partial Differential Equations**Plot the direction field:**First Order Linear Partial Differential Equations**Plot the direction field: t x**First Order Linear Partial Differential Equations**Plot the direction field: t x**First Order Linear Partial Differential Equations**Direction field: Through every point, a curve exists that is tangent to <a,b> everywhere. t x**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**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.**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:**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**First Order Linear Partial Differential Equations**SHORT INTERMEZZO**First Order Linear Partial Differential Equations**SHORT INTERMEZZO Parameterization of a line in 2 dimensions Parameter representation of a circle**First Order Linear Partial Differential Equations**Or in 3 dimensions Parameter representation of a helix**First Order Linear Partial Differential Equations**Or in 3 dimensions NOW BACK TO THE CHARACTERISTIC BASE CURVES Parameter representation of a helix**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**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**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:**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:**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:**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**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:**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).**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**First Order Linear Partial Differential Equations**The equations for the characteristic base were solved to get the base curves in the example: Solving**First Order Linear Partial Differential Equations**The equations for the characteristic base were solved to get the base curves in the example: Solving gives**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!**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).**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 .**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).**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.**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**First Order Linear Partial Differential Equations**• Consider with The corresponding PDE reads:**First Order Linear Partial Differential Equations**• Consider with • Parameterize this initial curve with parameter l:**First Order Linear Partial Differential Equations**• Consider with • Parameterize this initial curve with parameter l: • Solve the characteristic equations with these initial conditions.**First Order Linear Partial Differential Equations**• Consider with • The (parameterized) solution reads:**First Order Linear Partial Differential Equations**Visualize the solution for various values of l:**First Order Linear Partial Differential Equations**When all values of l and s are considered, we get the solution surface:**First Order Linear Partial Differential Equations**PDE: Initial condition:**First Order Linear Partial Differential Equations**PDE: Initial condition: Char eqns:**First Order Linear Partial Differential Equations**PDE: Initial condition: Char eqns: Parameterised initial condition:**First Order Linear Partial Differential Equations**Char eqns: Parameterized initial condition: Parameterized solution:**First Order Linear Partial Differential Equations**Char eqns: Initial condition: