# What is Numerical method: Definition and 42 Discussions

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.

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1. ### A Numerically solving a transport equation

I'm using a downwind'' approximation for the spatial derivative: \frac{\partial v}{\partial x}\approx -\frac{3}{2h}v_{j}+\frac{2}{h}v_{j-1}-\frac{1}{2h}v_{j-2} I'm using the usual approximation for the time derivative, I get the following for a stencil...
2. ### 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...
3. ### How to judge the singularity of a matrix in numerical method?

Summary: different methods give different results. They are not consistent. Summary: different methods give different results. They are not consistent. I use two different methods to detect whether a matrix is singular. The result of calculating the determinant of a 9-order square matrix is...
4. ### Mathematica Solving 2-D partial integro-differential equation

While reproducing a research paper, I came across the following equation, ∂f/∂t−(H(f)(∂f/∂x)=0 where [H(f)] is hilbert transform of 'f.' and f=f(x,t) and initial condition is f(x,0)=cos(x) and also has periodic boundary conditions given by F{H{f(x′,t)}}=i⋅sgn(k)F{f(x,t)}, where F(f(x,t) is...

9. ### How can I be sure of my numerical result?

Homework Statement In this problem you will do numerical computer calculations. A skydiver of mass 75.0 kg jumps out of a plane at an altitude of 30.0 km above the surface of the Earth. His parachute fails to open. Assume there is no horizontal motion and the initial velocity is zero. We...
10. ### I Generating polynomials for a multistep method

Hi, I'm struggling to understand how the generating polynomials work and are implemented in the difference equation for a general ODE y' = f(t,y) Difference Equation Generating polynomials "Coefficients are normalized either by a_k = 1 or sigma(1) = 1
11. ### A Numerical solution to SE - variational method, many electrons

Hi everyone, I am trying to find electron wavefunction of a system I am working in. Numerical method I choose is the Variational method (VM). This method is convenient to find the ground state of the system. More details are available here. Problem I have can be explained on a very simple...
12. ### Selecting a numerical method

Good Day Let's say I have developed a new method to extract, more efficiently (yes, "more efficiently" is ill-defined; but bear with me) the differential equations that describe a specific phenomena (please just assume it). So now I have a system of coupled second order differential...
13. ### Numerical methods that need a guess/approximate solutions

Hello everyone! I am currently playing with an old analog computer, which could solve time-dependent ODE/PDEs pretty fast, without time-stepping. But the problem with analog computer's solutions is that they are not very accurate. I am very curious that is there any numerical method/solver which...
14. ### How to compute the surface height based on normal vectors

Suppose I have already found the surface normal vectors to a set of points (x,y), how do I compute the surface height z(x,y)? Basically what I have are the normal vectors at each point (x,y) on a square grid. Then I calculate the vectors u = (x+1,y,z(x+1,y)) - (x,y,z(x,y)) and v =...
15. ### MHB Implementation of numerical method

Hello! (Smile) Consider the initial value problem $$\left\{\begin{matrix} y'(t)=f(t,y(t)) &, a \leq t \leq b \\ y(a)=y_0& \end{matrix}\right. (1)$$ I want to write a program that implements the following numerical method to solve $(1)$ \$\left\{\begin{matrix} y^{n+1}=y^n+h[\rho...
16. ### Algorithm for Numerical approximation to add data points

Hi, I am working on TDR (Time Domain Reflectometry). I send a 7GHz bandwidth fast rising edge (14ns) square wave into a coax. I get a return Signal. I have an ADC with 10Msamples/sec. I am using MPLAB IDE for coding the microcontroller. Now I would like to increase the Points on the...
17. ### Numerical method for wave equation

Hi, I am trying to plot a function subjected to a nonlinear wave equation. One of the method I found for solving the nonlinear schrodinger equation is the split step Fourier method. However I noticed that this method only works for a specific form of PDE where the equation has an analytic...
18. ### Numerical method to solve a spring system

Hello everyone, this is my first thread in this comunity. I explain my problem: I have a triangle mesh http://en.wikipedia.org/wiki/Triangle_mesh where at every vertex coincide a mass element and on every edge I add a spring. I would like simulate a deformation model. My idea derive...
19. ### Integration of 2 product. possible without numerical method?

Dear Forum : I hung up with a integration http://ppt.cc/mIpV Can it be deduced to a simpler form? The distribution of σ(E) is http://ppt.cc/-5Z5 The estimation width of x is 10MeV , height is 200mb. The distribution of dE/dx is http://ppt.cc/vcVU Is there a way to do some simple...
20. ### Numerical method to use on a system of second order nonlinear ODE's

I'm trying to create a java application that models the path of a double pendulum. To do so I have been attempting to use Lagrangian Mechanics to find the equation's of motion for the system. The problem is that I have never seen a set of equations like the one yielded by this method and need...
21. ### Numerical method for finding charge density on a conductor

I'm interested in working out an efficient and precise numerical method to find the charge density on a conductor with an axially symmetric shape that is essentially a deformed sphere. (I have in mind something like r(\theta)=\sum \beta_\ell P_\ell(\cos\theta), where P is a Legendre polynomial.)...
22. ### Numerical method: iteratation to converge on root

Hello! I'm having trouble understanding the method/reasoning behind finding the root of an equation though iterative convergence. x2 - 4x + 1 = 0 x2 = + 4x - 1 x = 4 - 1/x I can understand that once we input a 'root' the equation will equal be equal on both sides. (Due to the remainder...
23. ### Numerical method for a heat diffusion problem

Hello everyone! I hope you may be able to shed light on a recent problem that i have to deal with. I keep working on a numerical solution on a heat diffusion problem. It is a numerical solution for the PDE θT/θt=a*(θ^2T/θx^2 + θ^2Τ/θy^2), that describes the heat diffusion on a two...
24. ### How Can Modified Convergence Criteria Enhance Regula Falsi Method Efficiency?

Using ΔX=XH-XL as one of the convergence criteria in regula falsi may lead to infinite looping, e.g. f(x)=0.5x-2ln(x), with a bracket [7,10]. A better choice is to use the approximation error ΔX=|ΔXR|, i.e, the difference between the current estimate and the previous estimate of the root...
25. ### Nonuniform mesh with the numerical method of characteristics

Could someone please direct me to a good book/article that describes how to use a coarse and fine spatial mesh with the numerical method of characteristics? Thank you. Kate
26. ### Numerical method on heat diffusion problem

Hello there! I am solving a heat diffusion problem which is described by the differential equation: θΤ/θt=a*(θ^2Τ/ΘΤ^2). I will apply two diffrent temperatures on the ends of a metal rod. Solving the equation i conclude, according to the boundary conditions of course, at the function...
27. ### Help with the numerical method of characteristics

I have a system of first order pdes of the form dm/dt+f(m,n,a)dm/dx = g1(m,n,a), dn/dt+f(m,n,a)dn/dx = g2(m,n,a), da/dt+f(m,n,a)da/dx = g3(m,n,a). (those d's are partials) I want to solve them with the numerical MOC's so I put dx/dt = f(m,n,a), dm/dt = g1(m,n,a) etc and solve this system...
28. ### What is the best numerical method for

An equation of the form: \frac{\partial A}{\partial t}+\frac{\partial B}{\partial x}=C I am thinking Lax-Wendroff.
29. ### The order of error of a numerical method

Homework Statement Find the order of error of this numerical method: f''(x_i+\frac{h}{2})\simeq \frac{\Delta ^2f_i}{h^2} Homework Equations Taylor's theorem, Newton's interpolation polynomial and its error. The Attempt at a Solution I started from Taylor's expansion at x_i and...
30. ### I want to analise this problem by numerical method

Homework Statement y=(a*exp^b*x)/(c+d*exp^m*x) The Attempt at a Solution (1/y)=(c+d*exp^m*x)/(a*exp^b*x) y(new)=1/y y(new)=(c/(a*exp^b*x))+(d*exp^m*x)/(a*exp^b*x) i stopped here and i don't know how can i complete?
31. ### Numerical method to solve ODE boundary problem

can anyone provide a Numerical algorithm to solve -y'' (x) +f(x)y(x) = \lambda _{n} y(x) with the boundary condition y(0)=y(a)=0 here 'a' is a parameter introduced at hand inside the program and f(x) is also introduced by hand in the program i am more interested in getting...
32. ### How to Create a Bisection Method Program for Arbitrary Functions?

help me please "numerical method" please help me finish college assignment, I was given the task of creating programs numerical calculation method, the bisection method, the program c + + or Visual Basic, or any program language. last collection time Friday, 23 April 2010. please help me...
33. ### Finding fluid flow with Numerical Method

Homework Statement For fluid flow in pipes, friction is described by a dimensionless number, the Fanning friction factor (f). The Fanning friction factor is dependent on a number of parameters related to the size of the pipe and the fluid, which all can be represented by another dimensionless...
34. ### Solving tan x = x: A Numerical Method

Homework Statement tan x = x Homework Equations The Attempt at a Solution how do i solve this equation without using graphs? is there a numerical method ? thanks a lot!
35. ### Error Analysis in Numerical Methods for Solving Equations

Homework Statement a. Using equation x2 – a = 0, show that x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right) b. Given that Xn = √a + e, where e is a small error. Show that: x_{n+1}=a^{1/2}+\frac{1}{2}\left(\frac{e^2}{a^{1/2}+e}\right) c. If x0 = 4 and a = 7, where √7 = 2.646, find...
36. ### Numerical method needed for the following system of ODEs

I am required to solve the following system of ODEs numerically. Could someone suggest an appropriate methodology. These equations are phenomenological equations derived from irreversible thermodynamics. I have to solve for the flux terms given on the L.H.S. The coefficients of differentials on...
37. ### Root of Non-Linear Eqn (Numerical Method)

Hi, I have a non-linear function F: \Re^{3}\rightarrow\Re. I would like to find the roots of this equation numerically, since an explicit formula cannot be derived. As far as I am aware Newton's method can only be utilized when the domain and the range of the function are of the same...
38. ### Numerical method to solve a fourth order ODE

I've got the following ODE: K*w''''+c*w = q I can solve this with DSolve (from Matlab), but how can I solve this in a numerical way? Thanks!
39. ### Numerical method to solve high order ODEs.

here is a simplified version of my working equtions y''' = \frac{(y'' y+y' y) y + y'y''}{y' + y''} and 3 related boundary conditions, is there some hints to solve such equation numerically? ThX
40. ### Numerical Method Set of questions

Hi, I'm currently stuck with 2 questions: 1. Given that the negative root of the equation f(x) = x^{3} - 7(x) + 5 lies between a and a + 1 where a is an integer write down a value of a 2. Show that the equation e^{-x} = x^{2} has a root between x = 0.70 and 0.71 Thanks :)
41. ### Three body problem and numerical method

What is meant by " a numerical method to solve a three body problem?"
42. ### How Does the Explicit Method Solve 1-D Heat Conduction Problems?

I have been given the following problem as assignment: Find a numerical solution for the 1-D heat conduction (using the Explicit Method): \left\{\begin{array}U_{xx} = U_{t},\\ U(x,0) = \sin \pi x, \\ U(0,t) = U(1,t) = 0 Use h = 1, k = 0.005125 and M = 200. Can anyone help by giving...