\( \frac{\partial T}{\partial t} + u_x\frac{\partial T}{\partial x} + u_y\frac{\partial T}{\partial y} - \kappa \left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = 0 \)
\( u_x = 1-y^2, \; u_y = 0, \; \kappa = 1 \)
\( 0.1 \leq x \leq 2.1, \; -1.1 \leq y \leq 1.1, \; 0.1 \leq t \leq 20 \)
\( \frac{dT(y = -1.1)}{dy} = \frac{dT(y = 1.1)}{dy} = 0 , \; T(x = 0.1) = T(x = 2.1), \; \frac{dT(x = 0.1)}{dx} = \frac{dT(x = 2.1)}{dx} \)
\( \text{Initial Condition} = \frac{\exp{\left(-((x - 1.1)^2 + y^2)\right)}}{0.08} \)
Approximate Formula: \( T = \left( \mathcal{I}^{\tanh(\mathcal{I})^{\sqrt{t}}} \right) - \left( \operatorname{sech}\left( \mathcal{I} + \frac{t}{0.2 \cdot y} \right) \cdot \operatorname{sech}\left( x + \left( y + 2^{\mathcal{I}} \right) \right) \right), \quad \mathcal{I} = \frac{\exp{\left(-((x - 1.1)^2 + y^2)\right)}}{0.08} \)
MSE: 0.00379332
Approximate Formula: \( T = \left( \mathcal{I}^{\tanh(\mathcal{I})^{\sqrt{t}}} \right) - \left( \operatorname{sech}\left( \mathcal{I} + \frac{t}{0.2 \cdot y} \right) \cdot \operatorname{sech}\left( x + \left( y + 2^{\mathcal{I}} \right) \right) \right), \quad \mathcal{I} = \frac{\exp{\left(-((x - 1.1)^2 + y^2)\right)}}{0.08} \)
MSE: 0.00379332