# Tagged Questions

Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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### Implicit system differential equations

I came across a system of differential equations in the form: $\newcommand{\D}[1]{\frac{\mathrm{d}#1}{\mathrm{d}x}}$ \begin{align} f_1(x,y,z)\D{y}+f_2(x,y,z)\D{z}&=f_3(x,y,z),\\ f_4(x,y,z)\D{y}+...
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### sturm-liouville differential equations

I need method to find the values of $\lambda$ $$1.-y''+x^2 y=\lambda y$$ $$2. -y''+|x| y=\lambda y$$ $$3. -y''+(x^2 +x^4) y=\lambda y$$ with initial condition $y(0)=1,y'(0)=0$
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### How to prove coercivity

I have a problem in understanding how to prove if a function is positive or negative coercive. I understood the definition of coercivity, which is: $$\lim_{||x|| \to +\infty}f(x) = +\infty$$ However, ...
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### Finite element method - master element

What is supposed to be a quadratic master element with 3 degrees of freedom? I think I have to consider a 3D case, with x,y,z directions...is it? And I think it is quadratic because I have to go ...
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### Solution of ode system (morphogenesis) has one value once large once small

I solve Turing's morphogenesis with code available in following question: Solve Turing's morphogenesis with other method than Euler's The problem is: the pictures are nice, however there ...
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### Prove existence of divergent sequence in Newton's method

Given $f(x)=x^3-1$, how to prove that there exists a sequence of initial values $x_{0,1}>x_{0,2}>x_{0,3}>...$ where $x_{0,1}=0,x_{0,2}=-2^{1/3}$, such that the sequence produced by Newton's ...
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### Can $\int_0^1 \frac{1}{x} e^{-x} dx$ be integrated?

I have an integral with a singularity at $x = 0$. $$\int_0^1 \frac{1}{x} e^{-x} dx$$ It's not a removable singularity so is it possible to perform the integration? For example could some complex ...
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### using Euler's method to solve this question ($\frac{dv}{dt}=-kA$) [closed]

Suppose that a spherical droplet of liquid evaporates at a rate that is proportional to its surface area. dv/dt= -kA where V=volume (mm3), t =time (min), k =the evaporation rate (mm/min), and A =...
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### Example of “no analytical solution”

Is there a good test for no analytical solution? How can I learn the difference between equations that have an analytical solution and the ones that need numerical methods ("unsolvables" in analysis)?
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### Gauss Seidel - Finite Element Method

I am solving an equation using finite element method, and for that I have to use Gauss Seidel to invert a matrix. In Gauss Seidel I am using a "while" which breaks if the absolute error reaches the ...
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### polynomials/numerical analysis

Suppose that $n ≥ 1$. The function $f$ and its derivatives of order up to and including $2n + 1$ are continuous on $[a, b]$. The points $x_i, i = 0, 1, \ldots , n$ are distinct and lie in $[a, b]$. ...
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So I have an ODE that needs to be solved a few thousand times on MATLAB and am wondering what the most efficient method to use would be. I am changing a constant term each time. My ODE is of the form $... 0answers 45 views ### Find the steady-states of the system of differential equations using sympy (in python) and determine their local stabilities. The system is given by:$\frac{dx}{dt} = r x(1 - x) - \beta x y$,$\frac{dy}{dt} = \beta x y - \gamma y$. Analytically, I have found the Jacobian is given by:$J(x,y) = \begin{bmatrix} r(1 - ...
I want to build a finite-difference approximation of this derivative: $\frac{\partial^2T }{\partial x^2}$ There are given an error of approximation: $O(\Delta x^{4})$ and nodal values of function:\$ ...