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42 lines
1.4 KiB
Markdown
42 lines
1.4 KiB
Markdown
# Volterra Integral Differential Equation
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We consider the first-order Volterra type integro-differential equation on $[0, 5]$ (from [Lu et al. 2021](https://epubs.siam.org/doi/abs/10.1137/19M1274067)):
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$$
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\frac{d y}{d x}+y(x)=\int_{0}^{x} e^{t-x} y(t) d t, \quad y(0)=1
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$$
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with the ground truth $u=\exp(-x) \cosh x$.
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## 1D integral with Variable Limits
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The LHS is represented by
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```python
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exp_lhs = sc.ExpressionNode(expression=f.diff(x) + f, name='lhs')
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```
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The RHS has an integral with variable limits. Therefore, we introduce the class `Int1DNode`:
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```python
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fs = sp.Symbol('fs')
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exp_rhs = sc.Int1DNode(expression=sp.exp(s - x) * fs, var=s, lb=0, ub=x, expression_name='rhs',
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funs={'fs': {'eval': netnode,
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'input_map': {'x': 's'},
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'output_map': {'f': 'fs'}}},
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degree=10)
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```
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We map `f` and `x` to `fs` and `s` in the integral, respectively.
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The numerical integration is approximated by Gauss–Legendre quadrature with `degree=10`.
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The difference between the RHS and the LHS is presented by a `pde_op.opterator.Difference` node,
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```python
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diff = sc.Difference(T='lhs', S='rhs', dim=1, time=False)
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```
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which generates a node with
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- `input=(lhs,rhs)`;
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- `output=(difference_lhs_rhs,)`.
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The final result is shown as follows:
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See `examples/Volterra_IDE`. |