FredholmIntegralEquationToSturmLiouvilleform - crowlogic/arb4j GitHub Wiki
The process of converting an integral equation to a differential equation, particularly in the case of a Sturm-Liouville problem, involves a few steps. I'll describe it in a generalized form and then touch on the specifics of a translation-invariant kernel.
General Process
Given an integral equation of the form:
$$ \int_a^b K(s, t) f(t) dt = \lambda f(s) $$
where $K(s, t)$ is the kernel, $f(t)$ is the eigenfunction, and $\lambda$ is the eigenvalue.
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Differentiate Both Sides: If the kernel $K(s, t)$ depends on $s$ and $t$ in a way that allows for differentiation with respect to $s$, you differentiate both sides of the equation with respect to $s$, potentially multiple times if needed, to obtain derivatives of $f$ with respect to $s$.
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Apply Leibniz's Rule: When differentiating under the integral sign, you apply Leibniz's rule, which can simplify the kernel into a form involving delta functions or other simplifications, especially if the kernel is made up of absolute value functions or other piecewise definitions.
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Obtain a Differential Operator: The goal is to express the left-hand side as a differential operator acting on $f$. This often involves the kernel transforming into a differential expression involving the function $f$ and its derivatives $f', f'', \ldots$, along with potentially new functions $p(t), q(t), \ldots$ that are derived from the properties of the kernel.
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Formulate Sturm-Liouville Problem: The result is a differential equation of the form:
$$ \frac{d}{dt}\left[p(t)\frac{df}{dt}\right] + q(t)f = \lambda w(t)f $$
where $p(t)$, $q(t)$, and $w(t)$ are coefficients that depend on the original kernel and the differential operator is self-adjoint.
Translation-Invariant Kernel
For a translation-invariant kernel, which depends only on the difference $s - t$, denoted as $K(s - t)$, and is absolutely continuous and infinitely differentiable, the simplification is as follows:
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Kernel Form: The kernel can be expressed as $K(s - t)$, which simplifies the analysis since it only depends on one variable $u = s - t$.
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Fourier Transform: Such kernels are amenable to analysis via the Fourier transform. The Fourier transform of the kernel $\hat{K}(\omega)$ and the eigenfunction $\hat{f}(\omega)$ can be used to convert the integral equation into an algebraic equation.
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Simplified Differential Operator: Since the kernel is infinitely differentiable, its derivatives with respect to $s$ under the integral sign will not encounter discontinuities, allowing for a straightforward application of Leibniz's rule.
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Application of Fourier Transform: By taking the Fourier transform of both sides, the convolution theorem can be applied, which states that the Fourier transform of a convolution is the product of the Fourier transforms:
$$ \hat{K}(\omega) \cdot \hat{f}(\omega) = \lambda \hat{f}(\omega) $$
- Eigenvalue Problem: This yields a simple eigenvalue problem in the frequency domain, where the eigenvalues are related to the Fourier transform of the kernel:
$$ \hat{f}(\omega) = \frac{1}{\lambda} \hat{K}(\omega) \cdot \hat{f}(\omega) $$
The simplifications provided by the translation-invariant, absolutely continuous, and infinitely differentiable properties of the kernel allow for a more straightforward analysis and solution to the problem, often enabling solutions in closed form, especially when leveraging the properties of the Fourier transform.