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376376### Linear and Non-Linear Operators
377377
378- The linear and non-linear operators can be defined as:
378+ The linear and non-linear operators can be defined as follows :
379379
380- - ** Linear Operator ($ \text{opL} $)** :
381- $$
382- \text{opL} = \exp(D (k^2 + \gamma k^4) \Delta t)
383- $$
384-
385- - ** Non-Linear Operator ($ \text{opN} $)** :
386- $$
387- \text{opN} = \frac{\exp(D (k^2 + \gamma k^4) \Delta t) - 1}{k^2 + \gamma k^4} k^2
388- $$
380+ $$
381+ \begin{align}
382+ \text{opL} &= \exp(D (k^2 + \gamma k^4) \Delta t) \\
383+ \text{opN} &= \frac{\exp(D (k^2 + \gamma k^4) \Delta t) - 1}{k^2 + \gamma k^4} k^2
384+ \end{align}
385+ $$
389386
390387These operators are used to update the concentration field $c$ in the
391388Fourier domain efficiently, leveraging the FFT for computational efficiency.
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