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8. Выберите из представленных ниже разностных соотношений те, которые составляют разностную схему Саульева, аппроксимирующую дифференциальное уравнение:
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\(\displaystyle \frac{\partial u}{\partial t}=0{,}5\frac{\partial^2 u}{\partial x^2}-2u+t.\) |
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\(\displaystyle \frac{u_j^{n+1}-u_j^n}{\Delta t}=0{,}5\frac{u_{j+1}^n-(u_j^n+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+1}+\Delta t n,\\ \frac{u_j^{n+2}-u_j^{n+1}}{\Delta t}=0{,}5\frac{u_{j+1}^{n+2}-(u_j^{n+2}+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+2}+\Delta t(n+1).\) |
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\(\displaystyle \frac{u_j^{n+1}-u_j^n}{\Delta t}=0{,}25\frac{u_{j+1}^n-(u_j^n+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+1}+\Delta t(n+1),\\ \frac{u_j^{n+2}-u_j^{n+1}}{\Delta t}=0{,}25\frac{u_{j+1}^{n+2}-(u_j^{n+2}+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+1}+\Delta t(n+1).\) |
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\(\displaystyle \frac{u_j^{n+1}-u_j^n}{\Delta t}=0{,}5\frac{u_{j+1}^n-(u_j^n+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+1}+\Delta t(n+1).\) |
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\(\displaystyle \begin{aligned}&\frac{u_j^{n+1}-u_j^n}{\Delta t}=0{,}5\frac{u_{j+1}^n-(u_j^n+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+1}+\Delta t(n+1),\\&\frac{u_j^{n+2}-u_j^{n+1}}{\Delta t}=0{,}5\frac{u_{j+1}^{n+1}-(u_j^{n+1}+u_j^{n+2})+u_{j-1}^{n+2}}{h^2}-2u_j^{n+1}+\Delta t(n+1).\end{aligned}\) |
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\(\displaystyle \frac{u_j^{n+1}-u_j^n}{\Delta t}=0{,}5\frac{u_{j+1}^n-(u_j^n+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+1}+\Delta t(n+1),\\ \frac{u_j^{n+2}-u_j^{n+1}}{\Delta t}=0{,}5\frac{u_{j+1}^{n+2}-(u_j^{n+2}+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+1}+\Delta t(n+1).\) |
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\(\displaystyle \frac{u_j^{n+1}-u_j^n}{\Delta t}=0{,}5\frac{u_{j+1}^n-(u_j^n+u_j^{n+1})+u_{j-1}^{n+1}}{h^2}-2u_j^{n+1}+\Delta t(n+1),\\ \frac{u_j^{n+2}-u_j^{n+1}}{\Delta t}=0{,}5\frac{u_{j+1}^{n+2}-2u_j^{n+2}+u_{j-1}^{n+2}}{h^2}-2u_j^{n+1}+\Delta t(n+1).\) |
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