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5. Для численного решения дифференциального уравнения
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\(\displaystyle \frac{\partial u}{\partial t} + 2\,\frac{\partial u}{\partial x} = \frac{\partial^2 u}{\partial x^2} - 4ux^2 + e^{t+x^2}, \qquad x \in [0, 1], \quad t \in [0, 1]\) |
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требуется подобрать устойчивую разностную схему. Определите,
в каком случае построение тестовой задачи для данного уравнения выполнено без ошибок.
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\(\displaystyle \bar{u} = e^{t+x^2} \quad \Rightarrow \quad \frac{\partial \bar{u}}{\partial t} = e^{t+x^2}, \quad \frac{\partial \bar{u}}{\partial x} = 2xe^{t+x^2}, \quad \frac{\partial^2 \bar{u}}{\partial x^2} = 4x^2 e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + \varphi(t, x);\quad e^{t+x^2} + 4xe^{t+x^2} = 4x^2 e^{t+x^2} - 4x^2 e^{t+x^2} + e^{t+x^2} + \varphi(t, x) \quad \Rightarrow \quad \varphi(t, x) = 4x^2 e^{t+x^2} + 4xe^{t+x^2} - 4x^2 e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + 4x^2 e^{t+x^2} + 4xe^{t+x^2} - 4x^2 e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} e^{t+x^2}(1 + 4x - 4x^2);\quad \bar{u}(t = 0, x) = 0, \quad \bar{u}(t, x = 0) = 0, \quad \bar{u}(t, x = 1) = 1\) |
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\(\displaystyle \bar{u} = e^{t+x^2} \quad \Rightarrow \quad \frac{\partial \bar{u}}{\partial t} = e^{t+x^2}, \quad \frac{\partial \bar{u}}{\partial x} = 2xe^{t+x^2}, \quad \frac{\partial^2 \bar{u}}{\partial x^2} = 4x^2 e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + \varphi(t, x);\quad e^{t+x^2} + 4xe^{t+x^2} = 4x^2 e^{t+x^2} - 4x^2 e^{t+x^2} + e^{t+x^2} + \varphi(t, x) \quad \Rightarrow \quad \varphi(t, x) = 4x^2 e^{t+x^2} + 4xe^{t+x^2} - 4x^2 e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} e^{t+x^2}(1 + 4x - 4x^2);\quad \bar{u}(t = 0, x) = e^{x^2}, \quad \bar{u}(t, x = 0) = e^t, \quad \bar{u}(t, x = 1) = e^{t+1}\) |
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\(\displaystyle \bar{u} = e^{t+x^2} \quad \Rightarrow \quad \frac{\partial \bar{u}}{\partial t} = e^{t+x^2}, \quad \frac{\partial \bar{u}}{\partial x} = 2xe^{t+x^2}, \quad \frac{\partial^2 \bar{u}}{\partial x^2} = 4x^2 e^{t+x^2} + 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + \varphi(t, x);\quad e^{t+x^2} + 4xe^{t+x^2} = 4x^2 e^{t+x^2} + 2e^{t+x^2} - 4x^2 e^{t+x^2} + e^{t+x^2} + \varphi(t, x) \quad \Rightarrow \quad \varphi(t, x) = 4xe^{t+x^2} - 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + 4xe^{t+x^2} - 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} e^{t+x^2}(4x - 1);\quad \bar{u}(t = 0, x) = e^{x^2}, \quad \bar{u}(t, x = 0) = e^t, \quad \bar{u}(t, x = 1) = e^{t+1}\) |
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\(\displaystyle \bar{u} = e^{t+x^2} \quad \Rightarrow \quad \frac{\partial \bar{u}}{\partial t} = e^{t+x^2}, \quad \frac{\partial \bar{u}}{\partial x} = 2xe^{t+x^2}, \quad \frac{\partial^2 \bar{u}}{\partial x^2} = 4x^2 e^{t+x^2} + 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + \varphi(t, x);\quad e^{t+x^2} + 4xe^{t+x^2} = 4x^2 e^{t+x^2} + 2e^{t+x^2} - 4x^2 e^{t+x^2} + e^{t+x^2} + \varphi(t, x) \quad \Rightarrow \quad \varphi(t, x) = 4xe^{t+x^2} - 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + 4xe^{t+x^2} - 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} e^{t+x^2}(4x - 1);\quad \bar{u}(t = 0, x) = e^{x^2}, \quad \bar{u}(t, x = 0) = e^t, \quad \bar{u}(t, x = 1) = e^{t+1}\) |
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\(\displaystyle \bar{u} = e^{t+x^2} \quad \Rightarrow \quad \frac{\partial \bar{u}}{\partial t} = e^{t+x^2}, \quad \frac{\partial \bar{u}}{\partial x} = 2xe^{t+x^2}, \quad \frac{\partial^2 \bar{u}}{\partial x^2} = 4x^2 e^{t+x^2} + 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + \varphi(t, x);\quad e^{t+x^2} + 4xe^{t+x^2} = 4x^2 e^{t+x^2} + 2e^{t+x^2} - 4x^2 e^{t+x^2} + e^{t+x^2} + \Phi(t, x) \quad \Rightarrow \quad \varphi(t, x) = 4xe^{t+x^2} - 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2}(x - 2)^2;\quad \bar{u}(t = 0, x) = e^{x^2}, \quad \bar{u}(t, x = 0) = e^t, \quad \bar{u}(t, x = 1) = e^{t+1}\) |
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\(\displaystyle \bar{u} = e^{t+x^2} \quad \Rightarrow \quad \frac{\partial \bar{u}}{\partial t} = e^{t+x^2}, \quad \frac{\partial \bar{u}}{\partial x} = 2xe^{t+x^2}, \quad \frac{\partial^2 \bar{u}}{\partial x^2} = 4x^2 e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + \varphi(t, x);\quad e^{t+x^2} + 4xe^{t+x^2} = 4x^2 e^{t+x^2} - 4x^2 e^{t+x^2} + e^{t+x^2} + \varphi(t, x) \quad \Rightarrow \quad \varphi(t, x) = 4xe^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2}(4x + 1);\quad \bar{u}(t = 0, x) = e^{x^2}, \quad \bar{u}(t, x = 0) = e^t, \quad \bar{u}(t, x = 1) = e^{t+1}\) |
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\(\displaystyle \bar{u} = e^{t+x^2} \quad \Rightarrow \quad \frac{\partial \bar{u}}{\partial t} = e^{t+x^2}, \quad \frac{\partial \bar{u}}{\partial x} = 2xe^{t+x^2}, \quad \frac{\partial^2 \bar{u}}{\partial x^2} = 4x^2 e^{t+x^2} + 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + \varphi(t, x);\quad e^{t+x^2} + 4xe^{t+x^2} = 4x^2 e^{t+x^2} + 2e^{t+x^2} - 4\bar{u}x^2 e^{t+x^2} + e^{t+x^2} + \varphi(t, x) \quad \Rightarrow \quad \varphi(t, x) = 4x^2 + 4xe^{t+x^2} - 4x^2 e^{t+x^2} - 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} - 4\bar{u}x^2 + e^{t+x^2} + 4\bar{u}x^2 + 4xe^{t+x^2} - 4x^2 e^{t+x^2} - 2e^{t+x^2};\quad \frac{\partial \bar{u}}{\partial t} + 2\,\frac{\partial \bar{u}}{\partial x} = \frac{\partial^2 \bar{u}}{\partial x^2} e^{t+x^2}(x - 2)^2;\quad \bar{u}(t = 0, x) = e^{x^2}, \quad \bar{u}(t, x = 0) = e^t, \quad \bar{u}(t, x = 1) = e^{t+1}\) |
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