Questions tagged [perturbation-theory]
Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.
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Why is this approximate solution correct?
Consider the following differential equation
$$ y''=-y + \alpha y |y|^2, $$
where $y=y(x)$ is complex in general and $\alpha$ is a real constant such that the second term is small compared to $y$ ($||^...
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Question related to isolated eigenvalue of a Hermitian operators
This picture is from the paper of F. J. Narcowich "Narcowich, F.J., 1980. Analytic properties of the boundary of the numerical range. Indiana University Mathematics Journal, 29(1), pp.67-77."...
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Invertibility of the product of matrices when the norm is less than 1
I am reading through a book about numerical linear algebra. It is challenging but I understand the concepts after some research. However, there is one thing that I can't figure out and it's about the ...
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Expansion of $y=\sqrt{1+x+\frac{\varepsilon}{\varepsilon+x}}$
I'm reading "Introduction to Perturbation Methods", Second Edition, by Mark H. Holmes.
In ch 2.2.5 "Matching Revisit" it explains the approach to match the outer and boundary layer ...
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Leading order matching of $\epsilon x^py'' + y' + y = 0$
Question: The function $y(x)$ satisfies $$\epsilon x^py'' + y' + y = 0,$$ in $x\in [0,1]$, where $p<1$, subject to the boundary conditions $y(0) = 0$ and $y(1)=1$. Find the rescaling for the ...
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Nondimensionalizing Fourth Order Differential Equation for an Elastic Beam Under Tension
I am going through the textbook A First Look At Perturbation Theory 2nd ed. by James G. Simmonds and James E. Mann Jr.
Exercise 1.14 states: "An elastic beam of section modulus $EI$, resting on ...
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Derivative of Spectral Radius of Matrix $\exp(A(t))$
I am faced with the practical problem of solving a system
$$\rho(\exp(A(t))) = 1$$ numerically, where $\rho$ signifies the spectral radius of the matrix $A(t) = B+ \frac{C}{t},$ $t \in (0, \infty)$.
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Can we say anything about how $\delta x$ and $\delta y$ are related to each other?
I have an equation of the form $$\frac{x^2}{y} = F(r), $$ where F is a function of $r$. This equation has a solution $r(x,y)$. Suppose we perturb this solution to $r(x + \delta x, y + \delta y)$ for ...
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Asymptotic expansion of $I(\alpha;\epsilon) = \int_0^{\infty}\frac{dx}{(\epsilon^2+x^2)^{\alpha/2}(1+x)}$
Question: Evaluate the first two terms of as $\epsilon \to 0$ of $$I(\alpha;\epsilon) = \int_0^{\infty}\frac{dx}{(\epsilon^2+x^2)^{\alpha/2}(1+x)},$$ for $\alpha = \frac{1}{2}, 1, 2$, if $$C(\alpha) = ...
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Asymptotic expansion of $\int_0^{\pi/2}\frac{\sin^2\theta}{(1-m^2\sin^2\theta)^{1/2}}d\theta$
Question: Evaluate the first two terms of
$$I(m) = \int_0^{\pi/2}\frac{\sin^2\theta}{(1-m^2\sin^2\theta)^{1/2}}d\theta$$ as $m\to 1^-$.
My approach: Setting $m = 1-\epsilon$ and $k = \sin\theta$, we ...
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Asymptotic approximation of an integral using splitting range
Question: Evaluate the first two terms as $\epsilon\to 0$ of $$\mathcal{N}(z)=\int_z^1\frac{dx}{\sqrt{x^3+\epsilon}}$$ where $0\le z <1$.
My approach: First, let us calculate $\mathcal{N}(z=0)$. It ...
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Bound on number of positive roots of deformed polynomial
In the comments for the following linked question Descartes rule of sign with positive real exponents, the following was stated:
" The positive roots depend continuously on the exponents. This ...
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Duffing equation with non-linearity factor greater than unity
I have been trying to solve the following non-linear equation taking help from the book regarding perturbative technique by H. Nayfeh (chapter -4)
$$\ddot{x}+\frac{x}{(1-x^2)^2}=0$$ where $x<1$ ...
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Applying WKB Method for a Fourth Order Schrodinger Like Equation
I was trying to apply the WKB method to analyze the approximate eigenvalue condition for the Schrodinger like equation
\begin{equation}
\varepsilon^{4} y^{(4)} = (E-V(x))y, \quad y(\pm \infty) = 0, V(...
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Given a perturbation to a symmetric matrix with multiple zero eigenvalues, seeking perturbation to eigenvalues
Let $A$ be a real, symmetric $n \times n$ matrix with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ where at least two of the eigenvalues are zero. Let $V$ be a real, symmetric $n \times n$ ...
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