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2. Численное решение системы дифференциальных уравнений
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\(\displaystyle \frac{\partial u}{\partial t} = v\,\frac{\partial^2 u}{\partial x^2} - ku\,v^3, \qquad \frac{\partial v}{\partial t} = u\,\frac{\partial^2 v}{\partial x^2} - ku\,v^3, \qquad u = u(t, x), \quad v = v(t, x)\) |
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проводится с использованием неявной разностной схемы. Из представленных ниже разностных схем выберите ту,
которая записана без ошибок.
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\(\displaystyle \frac{u_j^{n+1} - u_j^n}{\Delta t} = v_j^{n+1}\,\frac{u_{j+1}^{n+1} - 2u_j^{n+1} + u_{j-1}^{n+1}}{h^2} - ku_j^{n+1}(v_j^{n+1})^3;\quad \frac{v_j^{n+1} - v_j^n}{\Delta t} = u_j^{n+1}\,\frac{v_{j+1}^{n+1} - 2v_j^{n+1} + v_{j-1}^{n+1}}{h^2} - ku_j^{n+1}(v_j^{n+1})^3.\) |
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\(\displaystyle \frac{u_j^{n+1} - u_j^n}{\Delta t} = v_j^{n+1}\,\frac{u_{j+1}^{n+1} - 2u_j^{n+1} + u_{j-1}^{n+1}}{h^2} - ku_j^{n+1}(v_j^n)^3;\quad \frac{v_j^{n+1} - v_j^n}{\Delta t} = u_j^{n+1}\,\frac{v_{j+1}^{n+1} - 2v_j^{n+1} + v_{j-1}^{n+1}}{h^2} - ku_j^n(v_j^n)^2 v_j^{n+1}.\) |
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\(\displaystyle \frac{u_j^{n+1} - u_j^n}{\Delta t} = v_j^n\,\frac{u_{j+1}^{n+1} - 2u_j^{n+1} + u_{j-1}^{n+1}}{h^2} - ku_j^{n+1}(v_j^n)^3;\quad \frac{v_j^{n+1} - v_j^n}{\Delta t} = u_j^{n+1}\,\frac{v_{j+1}^{n+1} - 2v_j^{n+1} + v_{j-1}^{n+1}}{h^2} - ku_j^n(v_j^n)^2 v_j^{n+1}.\) |
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\(\displaystyle \frac{u_j^{n+1} - u_j^n}{\Delta t} = \nu_j^n\,\frac{u_{j+1}^{n+1} - 2u_j^{n+1} + u_{j-1}^{n+1}}{h^2} - ku_j^n(v_j^n)^3;\quad \frac{v_j^{n+1} - v_j^n}{\Delta t} = u_j^n\,\frac{v_{j+1}^{n+1} - 2v_j^{n+1} + v_{j-1}^{n+1}}{h^2} - ku_j^n(v_j^n)^3.\) |
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\(\displaystyle \frac{u_j^{n+1} - u_j^n}{\Delta t} = \nu_j^n\,\frac{u_{j+1}^{n+1} - 2u_j^{n+1} + u_{j-1}^{n+1}}{h^2} - ku_j^{n+1}(v_j^n)^3;\quad \frac{v_j^{n+1} - v_j^n}{\Delta t} = u_j^n\,\frac{v_{j+1}^{n+1} - 2v_j^{n+1} + v_{j-1}^{n+1}}{h^2} - ku_j^n(v_j^n)^2 v_j^{n+1}.\) |
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\(\displaystyle \frac{u_j^{n+1} - u_j^n}{\Delta t} = \nu_j^n\,\frac{u_{j+1}^{n+1} - 2u_j^{n+1} + u_{j-1}^{n+1}}{h^2} - ku_j^{n+1}(v_j^n)^3;\quad \frac{v_j^{n+1} - v_j^n}{\Delta t} = u_j^n\,\frac{v_{j+1}^{n+1} - 2v_j^{n+1} + v_{j-1}^{n+1}}{h^2} - ku_j^n(v_j^{n+1})^3.\) |
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