Question #124933
solve f(x, u_x,u_z)=g(y,u_y,u_z) by Jacobi method.
1
Expert's answer
2020-07-05T17:34:22-0400

Let f(x,y,z,ux,uy,uz)=0f(x,y,z,u_x,u_y,u_z) = 0 (1)

The Fundamental idea of the Jacobi method is to introduce two first order PDE involving two arbitrary constants a and b of the following form,

h1(x,y,z,ux,uy,uz,a)=0h_1(x,y,z,u_x,u_y,u_z,a) = 0 (2)

h2(x,y,z,ux,uy,uz,b)=0h_2(x,y,z,u_x,u_y,u_z,b) = 0 (3)

such that (f,h1,h2)(ux,uy,uz)0\frac{\partial(f,h_1,h_2)}{\partial(u_x,u_y,u_z)} \neq 0 (4)


and

Equations (1),(2) and (3) can be solved for ux,uy,uzu_x,u_y,u_z

and equation du=uxdx+uydy+uzdzdu = u_xdx + u_ydy+u_zdz is integrable


If h1=0h_1 = 0 and h2=0h_2=0 are compitable withf=0f =0 then h1,h2h_1,h_2 satisfy,

(f,h)(x,ux)+(f,h)(y,uy)+(f,h)(z,uz)=0\frac{\partial(f,h)}{\partial(x,u_x)}+\frac{\partial(f,h)}{\partial(y,u_y)}+\frac{\partial(f,h)}{\partial(z,u_z)} =0 for h=hih = h_i, i =1,2

Last equation will lead to the PDE of the form,

fuxhx+fuyhy+fuzhzfxhuxfyhuyfzhuz=0f_{u_x}\frac{\partial h}{\partial x}+f_{u_y}\frac{\partial h}{\partial y}+f_{u_z}\frac{\partial h}{\partial z} - f_x\frac{\partial h}{\partial u_x}- f_y\frac{\partial h}{\partial u_y}- f_z\frac{\partial h}{\partial u_z} = 0

It's auxiliary equation will be,

dxfux=dyfuy=dzfuz=duxfx=duyfy=duzfz\frac{dx}{f_{u_x}}=\frac{dy}{f_{u_y}}=\frac{dz}{f_{u_z}} = \frac{du_x}{-f_x}= \frac{du_y}{-f_y}= \frac{du_z}{-f_z}

After getting this we need to those fractions which will lead to the solution.


Since given equation is

f(x,ux,uz)=g(y,uy,uz)f(x,u_x,u_z) = g(y,u_y,u_z)

it can be simplified as

f(x,ux,uz)g(y,uy,uz)=0f(x,u_x,u_z) - g(y,u_y,u_z) = 0

or it can be written as

F(x,y,ux,uy,uz)=0F(x,y,u_x,u_y,u_z) = 0

Then auxiliary equation will be

dxFux=dyFuy=dxFuz=duxFx=duyFy\frac{dx}{F_{u_x}}=\frac{dy}{F_{u_y}} =\frac{dx}{F_{u_z}}= \frac{du_x}{-F_x}= \frac{du_y}{-F_y}

Solve it to obtain the solution.


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