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I am interested in how to select interesting yet reasonable problems for students to work on, either at Honours (that is, a research-based single year immediately after a degree) or PhD.
By this I mean a problem that is unsolved but for which there is a good chance that a student can solve it e...
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The usage of tag-removed was two-fold:
as an 'empty' tag for massively off-topic and mistagged questions.
as a tool to 'delete' tags via merging them into it.
Due to the second use-case numerous legitimate questions also got this tag. I think it could make sense to go over this list and reta...
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I have this system of $n$ non-linear equations in $n$ unknowns, arising out of my research problem. Given that $x_0=1$, I have to solve for $(x_1,x_2,\ldots,x_n).$
$$\sum_{i=0}^n x_i^2+2\sum_{j=1}^n\sum_{i=0}^{n-j}x_ix_{i+j}=1$$
$$\sum_{j=1}^n\sum_{i=0}^{n-j}~j^2~x_ix_{i+j}=0$$
$$\sum_{j=1}^n\sum...
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I am looking for an exact solution for the following special case of Chini Equation with $2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+$,
$$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$
I have tried to approach this using multiple methods and substitutions, but none has gotten me far. I know tha...
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Would anybody be able to share a Mathematica/Matlab/other script for calculating Onsager's exact solution for the magnetisation of the 2d Ising model? I would be most grateful of one in order to test my MC simulations of the system.
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I am looking for an exact solution to equation:
$w''=aw^{-1/3}+b[f(y)]w^{-5/3}$, where $w=w(y), f(y)$ - arbitrary function (in this case $y^n$ with arbitrary $n$); $a,b$ - constants.
Of course I can solve it using numerical methods for certain initial conditions, but I was looking for a exact s...
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The nonlinear pde
$$
\partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0
$$
has the exact solution
$$
\phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i)
$$
with $\mu$ and $\varphi$ two integration constants and sn the snoidal Jacobi function, provided the...
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It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features.
My question: Which Ansätze do you know to solve PDEs with Wavelets? Are these solution methods actually superior to the classical Ansätz...
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I have this system of $n$ non-linear equations in $n$ unknowns, arising out of my research problem. Given that $x_0=1$, I have to solve for $(x_1,x_2,\ldots,x_n).$
$$\sum_{i=0}^n x_i^2+2\sum_{j=1}^n\sum_{i=0}^{n-j}x_ix_{i+j}=1$$
$$\sum_{j=1}^n\sum_{i=0}^{n-j}~j^2~x_ix_{i+j}=0$$
$$\sum_{j=1}^n\sum...
![MathOverflow](http://cdn.sstatic.net/Sites/mathoverflow/Img/apple-touch-icon.png)
I am looking for an exact solution for the following special case of Chini Equation with $2\geq a > 1 > b > 0, x, y \in \mathbb{R}^+$,
$$\frac{dy}{dx} = 1 + \frac{a}{y} + \frac{b}{x}$$
I have tried to approach this using multiple methods and substitutions, but none has gotten me far. I know tha...
![MathOverflow](http://cdn.sstatic.net/Sites/mathoverflow/Img/apple-touch-icon.png)
I am looking for an exact solution to equation:
$w''=aw^{-1/3}+b[f(y)]w^{-5/3}$, where $w=w(y), f(y)$ - arbitrary function (in this case $y^n$ with arbitrary $n$); $a,b$ - constants.
Of course I can solve it using numerical methods for certain initial conditions, but I was looking for a exact s...
![MathOverflow](http://cdn.sstatic.net/Sites/mathoverflow/Img/apple-touch-icon.png)
The nonlinear pde
$$
\partial_t^2\phi-\partial_x^2\phi+\lambda\phi^3=0
$$
has the exact solution
$$
\phi(x,t)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p_0t-p\cdot x+\varphi,i)
$$
with $\mu$ and $\varphi$ two integration constants and sn the snoidal Jacobi function, provided the...
![MathOverflow](http://cdn.sstatic.net/Sites/mathoverflow/Img/apple-touch-icon.png)
It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features.
My question: Which Ansätze do you know to solve PDEs with Wavelets? Are these solution methods actually superior to the classical Ansätz...
![MathOverflow](http://cdn.sstatic.net/Sites/mathoverflow/Img/apple-touch-icon.png)
In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important:
$$\bigtriangleup F(x):=F(x+1)-F(x)=f(x) \quad\quad(1),$$
where $\bigtriangleup$ is the forward difference operato...
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