1 Quasi-Linear Partial Differential Equations

1 Quasi-Linear Partial Differential EquationsDefinition th order Partial differential equationis an equationinvolving the firstnpartial derivatives ofu,F(x,y,..,u, u x, u y,.., nu xn,..) = first-order two variablesx,yis an equation of typea(x,y) u x+b(x,y) u y=c(x,y)u(x,y).We will be able to solve Equations of this form; in fact of a slightly moregeneral form, so calledquasi- linear :a(x,y,u) u x+b(x,y,u) u y=c(x,y,u).2 SolutionDefine a curve in thex,y,uspace as followsr(t) = (x(t),y(t),z(t)) satisfyingdrdt= (a,b,c)with initial conditionr(0) on the surfacez=u(x,y).From Picard s theorem, assuming thata,bandcare well-behaved functionsofx,yandu, we know that there is a unique we can compare this curve with the curveu=u(x(t),y(t)).dudt= u xdxdt+ u ydydt=a u x+b u y=c=dzdtand since both curves start at the same point, we must haveu= course, this only gives a single curve on the surface, but if we are specifieda whole line of initial points of the typeu(xs,ys) =usthen we can solve theequation r= (a,b,c) with initial conditionsx(0) =xs,y(0) =ysandu(0) =usto give a collection of lines all lying on the surface we getu=u(t,s).

1 Quasi-Linear Partial Differential Equations Definition 1.1 An n’th order partial differential equation is an equation involving the first n partial derivatives of u,

Loading..

Tags:

  Linear, Partial, Linear partial

Information

Related search queries