\( \frac{\partial T}{\partial t} + u_x\frac{\partial T}{\partial x} + u_y\frac{\partial T}{\partial y} - \kappa \left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = 0 \)
\( u_x = \sin(4y), \; u_y = \cos(4x), \; \kappa = 1 \)
\( 0.1 \leq x \leq 2\pi, \; 0.1 \leq y \leq 2\pi, \; 0.1 \leq t \leq 20 \)
\( T(x = 0.1) = T(x = 2\pi), \; \frac{dT(x = 0.1)}{dx} = \frac{dT(x = 2\pi)}{dx}, \; T(y = 0.1) = T(y = 2\pi), \; \frac{dT(y = 0.1)}{dy} = \frac{dT(y = 2\pi)}{dy} \)
\( \text{Initial Condition} = \frac{\exp{\left(-((x - \pi)^2 + (y - \pi)^2)\right)}}{0.08} \)
Approximate Formula: \( T = \mathcal{I}^{\frac{0.2^{t}}{\text{sech}(\log(\log(\pi))) + 0.2 }} \approx \mathcal{I}^{0.2^t / 1.191}, \quad \mathcal{I} = \frac{\exp{\left(-((x - \pi)^2 + (y - \pi)^2)\right)}}{0.08} \)
MSE = 0.689072
Approximate Formula: \( T = \mathcal{I}^{\frac{0.2^{t}}{\text{sech}(\log(\log(\pi))) + 0.2 }} \approx \mathcal{I}^{0.2^t / 1.191}, \quad \mathcal{I} = \frac{\exp{\left(-((x - \pi)^2 + (y - \pi)^2)\right)}}{0.08} \)
MSE = 0.689072
Approximate Formula: \( T = \mathcal{I} + \operatorname{sech}\left( \frac{1/\pi + x + y}{20t^2}\right), \quad \mathcal{I} = \frac{\exp{\left(-((x - \pi)^2 + (y - \pi)^2)\right)}}{0.08} \)
MSE = 0.0117411
Approximate Formula: \( T = \mathcal{I} + \operatorname{sech}\left( \frac{1/\pi + x + y}{20t^2}\right), \quad \mathcal{I} = \frac{\exp{\left(-((x - \pi)^2 + (y - \pi)^2)\right)}}{0.08} \)
MSE = 0.0117411
