# What is Numerical calculation: Definition and 13 Discussions

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis naturally finds application in all fields of engineering and the physical sciences, but in the 21st century also the life sciences, social sciences, medicine, business and even the arts have adopted elements of scientific computations. The growth in computing power has revolutionized the use of realistic mathematical models in science and engineering, and subtle numerical analysis is required to implement these detailed models of the world. For example, ordinary differential equations appear in celestial mechanics (predicting the motions of planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.
Before the advent of modern computers, numerical methods often depended on hand interpolation formulas applied to data from large printed tables. Since the mid 20th century, computers calculate the required functions instead, but many of the same formulas nevertheless continue to be used as part of the software algorithms.The numerical point of view goes back to the earliest mathematical writings. A tablet from the Yale Babylonian Collection (YBC 7289), gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square.
Numerical analysis continues this long tradition: rather than exact symbolic answers, which can only be applied to real-world measurements by translation into digits, it gives approximate solutions within specified error bounds.

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1. ### A Solve PDE w/ Comsol 5.3: Numerical Solution & Time Evolution

What is the best way to solve numerically the following equation using Comsol 5.3. ##\frac{\partial T}{\partial t}=\frac{\partial ^2T}{\partial x^2}+\text{St}\left[1+\left(\frac{\partial T}{\partial x}\right)_{x=0}\right]\frac{\partial T}{\partial x}## ##T(0,t)=1## ##T(\infty ,t)=0##...
2. ### MATLAB Matlab's numeric solution to det of Matrix is incorrect

Hello everyone, I have an equation derived as the det of a matrix, which I have solved in Mathematica 11 with Findroot and verified with the respective PhD Thesis' data. However, I now try to get more accustomed to Matlab (R2017b) and hence I tried to reproduce the problem and find its solution...
3. ### A Numerical method to Lippman-Schwinger equation

There is a question that puzzle me when I apply numerical method to principal value integral. Let me descibe it. We make use of the fact that the integral ##\int_0^\infty \frac{dk}{k^2-k_0^2}## vanishes, namely, $$\int_0^\infty \frac{dk}{k^2-k_0^2} = 0 .$$ We use this formula to express a...
4. ### I Numerical Calculation of Hamiltonian Overlaps

Hi all, I am currently reading through this paper: https://iopscience.iop.org/article/10.1088/1367-2630/10/4/045030 and would like to reproduce their results for N=5. My roadblock is with (9), which models the classical motion of the system. Now symbolically finding the eigenstates of the matrix...
5. ### How Can Physics Forums Help Me with My Research Paper?

It is very amazing help8ng and encouraging forum. I am a research student of MS PHYSICS at COMSATS UNIVERSITY ISLAMABAD. I hope this firum will be much helpful to help me finding some tedious numerical solutions of a research paper.
6. ### Unexpected result in the numerical calculation - (invalid value encountered)

from numpy import log as ln z = 3 k = 2 x = 1 - ln(1 + z) / ln(1 + k) y = 1/5 print("The x=", x) Q = x**y print(Q) The result is The x= -0.26185950714291484 c:\Users\-\Desktop\... RuntimeWarning: invalid value encountered in double_scalars Q =...
7. ### Calculating eigenvectors/values from Hamiltonian

Homework Statement I've constructed a 3D grid of n points in each direction (x, y, z; cube) and calculated the potential at each point. For reference, the potential roughly looks like the harmonic oscillator: V≈r2+V0, referenced from the center of the cube. I'm then constructing the Hamiltonian...
8. ### A Numerical solution for a two dimensional 3rd order nonlinear diff. eq.

I'm a bit lost in all the numerous methods for solving differential equations and I would be very grateful if someone could point me to some direction. I want to solve the following boundary conditioned differential equation: a_1+a_2\nabla f(x,y)+a_3\nabla f(x,y)\cdot \nabla^2...
9. ### I How Can I Graph Hubble Term vs. Inflaton Using Mathematica?

From cosmology, ##H^2 = \frac{ρ}{3M_p^2} = \frac{1}{3M_p^2}(½\dot φ^2 + ½m^2φ^2)## Suppose ##V(φ) = ½m^2φ^2## where ##ρ## = density ##M_p## = Planck mass I want to graph ##H## vs. ##φ## but there is a ##\dot φ## and I know this is a differential equation, can somebody help me what to do here?
10. ### Numerical calculation of tunneling probability

Hi, I am trying to numerically calculate the tunneling probability for a wave function |psi> as a function of x. I have a double well potential. The wave function is initially in the left well. I do not know what exactly I have to find to show tunneling: the probability of being on the left well...
11. ### Numerical calculation about curve length

I think I have returned all my math back to teachers without any refund. y=f(x); h=xb-xa, which is very small. My Q is to calculate curve length rather than area numerically. But let me use area as example to show you what i want. to calculate area between xa to xb, we have 2...
12. ### Numerical Calculation of \Gamma = 2.354 \sigma

For starters... What does "data reduction" mean? The book starts throwing around this term (which is also part of the book's title) without even defining it. homework question: Show by numerical calculation that, for the Gaussian probability distribution, the full-width at half maximum...
13. ### On numerical calculation of Lift Force via Potential Flow

Imagine: I want to compute numerically a POTENTIAL STEADY and INCOMPRESSIBLE flow over an airfoil. The set up of the problem is: \nabla^2\phi=0 \nabla\phi \cdot \overline{n}\big)_{x=surface}=0 no normal velocity component on the airfoil surface. \nabla \phi=\overline{U_\infty} as...