The following differential schemes are used to solve the given problem:
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First Scheme: Combining equations (18), (21), and (22), we derive a differential scheme for solving problem (7). The general form is given as follows:
[
u^{n+1} = u^n + \Delta t \left( -\left( \frac{d^2 u}{dx^2} \right)^n \right)
]
In this case, the value of ( u_0 ) at layer ( j=0 ) is known. Using the above scheme, we can compute approximate values for nodes in the first layer ( j=1 ), then continue calculating successively for each layer.
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Classical Implicit Scheme: Rearranging equation (19) and combining with equations (21) and (22), we arrive at the following implicit differential scheme:
[
u^{n+1} = u^n + \Delta t \left( -\left( \frac{d^2 u}{dx^2} \right)^n + f(x, t) \right)
]
In this implicit scheme, though ( u_0 ) at the 0th layer is known, the calculation of values for subsequent layers ( j \geq 1 ) cannot be done directly. Hence, this scheme is termed the classical implicit format.
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Dufort-Frankel Scheme: The Dufort-Frankel scheme is a three-layer explicit scheme derived by combining equations (24), (25), and (26). Its specific form is:
[
u^{n+1} = u^n + \Delta t \left( \frac{1}{2} \left( \frac{d^2 u}{dx^2} \right)^n \right)
]
In this scheme, the value ( u_0 ) at the 0th layer is determined by the initial condition, and then values for subsequent layers are computed iteratively, starting from the 1st layer using a two-layer format.
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Hyperbolic Equation and Differential Solutions: For the second-order wave equation:
[
\frac{\partial^2 u}{\partial x^2} = a^2 \frac{\partial^2 u}{\partial t^2}
]
we define:
[
v = \frac{\partial u}{\partial x}, \quad \frac{\partial v}{\partial t} = \frac{\partial u}{\partial t}
]
This transforms the equation into a first-order linear system of hyperbolic equations."