# How to treat non-linear term in finite difference solution of $T''_x+T''_y+aT^2=0$?

Can we linearize $T^2$ When solving $T''_x+T''_y+aT^2=0$ by finite difference?

I solved $T''_x+T''_y=0$ in Matlab using a finite difference explicit scheme. But when there is a source term, I come up with a system of nonlinear algebraic equations and I can't solve it anymore.

Is there a better method for solving nonlinear equations without linearizing them?

• The one way I know of is to use iterative linear solvers, then the square term is made up of a component of the unknown solution you are searching for and a known component. So the equation becomes something like T''(x,n+1) + T''(y,n+1) + a T(n + 1) T(n) = 0, where n represents an iteration number. – user3209427 Dec 24 '17 at 13:34
• If you dont want to solve it exactly but approximately, you can make the following substitution: $T=T_0+/delta T$ With $/delta T$ small. Quadratic terms are then ommitted, and you can solve a linear equation. This only works if $/delta T/T_0$ is small enough. Check it after your calculation. – HBR Dec 24 '17 at 15:54