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Math Help - Quasi-Linear

  1. #1
    Junior Member sweet's Avatar
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    Quasi-Linear

    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 !!!
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  2. #2
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    Let u_1 \mbox{ and }u_2 solve a PDE, if it happens that c_1u_1+c_2u_2, i.e. any linear combination is a solution, then we say the PDE is linear.

    For example, consider the wave equation,
    \frac{\partial ^2 u}{\partial t^2} - \frac{\partial ^2 u}{\partial x^2} = 0.
    Confirm that if u_1(x,t)\mbox{ and }u_2(x,t) are solutions then c_1u_1(x,t)+c_2u_2(x,t) is a solution as well.

    Another way to think of linear is that u, the solution, is just like what linear means in an ordinary differencial equation. So u_{xx}+txu_x=0 is linear. But u_{xx}^2+txu_x=0 is not, because by analogy we never have (y'')^2+x^2y'=0 being considered linear ODE.

    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.
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  3. #3
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    2- How we can solve Quasi-Linear ??
    A general 2nd order Quasilinear equation is,
    A(x,y)\frac{\partial ^2 u}{\partial x^2}+B(x,y)\frac{\partial ^2 u}{\partial x \partial y} +C(x,y)\frac{\partial ^2 u}{\partial y^2} = F(x,y,u,u_x,u_y)

    Depending on the sign of B^2-4AC: 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.
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  4. #4
    Junior Member sweet's Avatar
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    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.
    really iwas need it , thank u

    but how can i solv quasi-linear of the first order..?
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  5. #5
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    Quote Originally Posted by sweet View Post
    but how can i solv quasi-linear of the first order..?
    Here is Lagrange's method from 1772 (older than America)

    Consider the equation,
    P(x,y,z)\frac{\partial z}{\partial x}+Q(x,y,z)\frac{\partial z}{\partial y}=R(x,y,z).

    Note, this is not a linear equation because P,Q,R contain z, 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),
    \frac{dx}{P} = \frac{dy}{Q} = \frac{dz}{R}.

    Let f(x,y,z)=c_1 \mbox{ and }g(x,y,z)=c_2 solve the differencial above (not the actual equation). Then the solution to the actual PDE is given by F(f,g)=0.
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  6. #6
    Junior Member sweet's Avatar
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    thank u

    do u have a paper of the Lagrange's method
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  7. #7
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    Quote Originally Posted by sweet View Post
    do u have a paper of the Lagrange's method
    Yes, I have the orginal paper. I am a direct descendent of Lagrange.
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