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I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation. $$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$ Now if the volatility is constant , then its the linear form. and if the volatility is variable, then its the nonlinear form ? Please give reference too with the answer if possible.

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The linear/non-linear classification is concerned about the dependent variables, and its derivatives. To verify whether the equation is linear, you should be checking that the equation is linear in each of these variables, and the coefficients of these are functions of the independent variables (t and x in your example).

In your example, the dependent variables and its derivative are $u$, $u_t$ and $u_{xx}$. As the equation is linear in all of them, the coefficient don't depend on u or its derivatives, and there are no cross terms (e.g., $u \, u_x$), so the equation is linear. The criteria you have listed - constant vs variable volatility - will help differentiate whether the equation is constant-coefficent. On the other hand, if volatility depends on u or its derivatives, then that would make it non-linear.

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