hi all...
i have 2 Q:
1-what is the diffrent between Quasi-Linear pde and Linear pde ?
2- How we can solve Quasi-Linear ??
any body can help !!!
Let solve a PDE, if it happens that , i.e. any linear combination is a solution, then we say the PDE is linear.
For example, consider the wave equation,
.
Confirm that if are solutions then is a solution as well.
Another way to think of linear is that , the solution, is just like what linear means in an ordinary differencial equation. So is linear. But is not, because by analogy we never have being considered linear ODE.
Quasi-linear means (this is how I understand it) that the highest order terms are linear.
For example,
Is not linear but it is quasi-linear. Because the highest order terms are linear.
A general 2nd order Quasilinear equation is,2- How we can solve Quasi-Linear ??
Depending on the sign of : positive, negative, zero. We have 3 cases: hyperbolic, elliptic, parabolic. Each one can be simplified by a method knows as charachteristics curves. When properly reduced by appropriate substitutions we get the "canonical form". From there we use certain PDE methods to solve that.
really iwas need it , thank u
Quasi-linear means (this is how I understand it) that the highest order terms are linear.
For example,
u_{xx}+\cos(x+y)u_{yy}+u_{xy}=e^{x+y} u_x\cdot u_y^2
Is not linear but it is quasi-linear. Because the highest order terms u_{xx},u_{yy},u_{xy} are linear.
but how can i solv quasi-linear of the first order..?
Here is Lagrange's method from 1772 (older than America)
Consider the equation,
.
Note, this is not a linear equation because contain , the solution, in them but still we can solve it.
We do this by solving, (I do not like using differencials but here they play a nice role),
.
Let solve the differencial above (not the actual equation). Then the solution to the actual PDE is given by .