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2. Выберите из представленных ниже разностных соотношений те,
которые составляют схему расщепления, аппроксимирующую дифференциальное уравнение:
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\(\displaystyle \frac{\partial u}{\partial t} = 5\frac{\partial^2 u}{\partial x^2} + 4\frac{\partial^2 u}{\partial y^2} - 9u.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t} = 5\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} - 9u_{j,k}^{n+1/2}\\ \frac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 4\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2}.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t} = 5\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} - 9u_{j,k}^{n+1/2},\\ \frac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t} = 4\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2}.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1} - u_{j,k}^n}{\Delta t} = 5\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} + 4\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2} - 9u_{j,k}^{n+1/2}\\ \frac{u_{j,k}^{n+1} - u_{j,k}^n}{\Delta t} = 5\frac{u_{j+1,k}^{n+1} - 2u_{j,k}^{n+1} + u_{j-1,k}^{n+1}}{h_x^2} + 4\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2}.\) |
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\(\displaystyle \frac{u_{j,k}^{n+1/2} - u_{j,k}^n}{\Delta t/2} = 5\frac{u_{j+1,k}^{n+1/2} - 2u_{j,k}^{n+1/2} + u_{j-1,k}^{n+1/2}}{h_x^2} - 9u_{j,k}^{n+1/2}\\ \frac{u_{j,k}^{n+1} - u_{j,k}^{n+1/2}}{\Delta t/2} = 4\frac{u_{j,k+1}^{n+1} - 2u_{j,k}^{n+1} + u_{j,k-1}^{n+1}}{h_y^2}.\) |
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