Example of quasi linear differential equation
Different Examples for Quasilinear Partial Differential Equations? Greating. Please, I want some Examples with solution if exists. More examples, including Mathematica notebooks can be foundExample 1: The equation @2u @x 2. a(x; y) @2u @y 2u 0 is a second order linear partial di erential equation. However, the following equation @u @x @2u @x2. @u @y @2u @y2. u2 0 is a second order quasilinear partial di erential equation. example of quasi linear differential equation
However, the method of characteristics can be applied to a form of nonlinear PDE. 1. 1 Trac ow. Ref: MyintU& Debnath 12. 6 Consider the idealized ow of trac along a onelane highway. Let (x, t) be the trac density at (x, t).
3 Example 2 As the second example we choose the following differential equation @v @t v @v @x v: (16) This has a trivial solution, and we directly solve it by the application of separation method. The answer to this equation is v (x) et; (17) where and are constants of integration. 1 BASIC CONCEPTS 7 order A rst order partial di erential equation is called quasilinear if it can be written in the form a(x; y; u)u. x b(x; y; u)u. y c(x; y; u): (1. 3) If a(x; y; u) (x; y) and b(x; y; u) (x; y) then (1. 3) is called semilinear. If futhermore, c(x; y; u) (x; y)u (x; y) then (1. 3) is called linear.example of quasi linear differential equation We will be able to solve equations of this form; in fact of a slightly more general form, so called quasilinear: a(x, y, u) u x b(x, y, u) u y c(x, y, u). Dene a curve in the x, y, u space as follows r(t) (x(t), y(t), z(t)) satisfying dr dt (a, b, c) with initial condition r(0) on the surface z u(x, y).