What is Initial conditions: Definition and 133 Discussions

In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n (that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time.
In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons.

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  1. U

    Why heat PDE solution does not fully satisfy initial conditions?

    Hi, I am solving heat equation with internal heat sources both numerically and analytically. My graphs are nearly identical but! analytical one have problem at the beginning and at the end for my domain. Many people have used the same technique to solve it analytically and they got good answers...
  2. Z

    Mathematica Mathematica bug? (Solving PDEs when the initial conditions contain a derivative)

    hello I own mathematica 10.02 it is virtually impossible to solve PDE's ,even with NDSolve,if the initial conditions contain a derivative I write Derivative[1,0] [0,x] == f[x] I mean the first t derivative of u[t,x] for x at t=0 is f[x] I own a book based on Mathematica 10.3 Even if a...
  3. A

    Solving a first order differential equation with initial conditions

    Hello! Consider this ODE; $$ x' = sin(t) (x+2) $$ with initial conditions x(0) = 1; Now I've solved it and according to wolfram alpha it is correct (I got the homogenous and the particular solution) $$ x = c * e^{-cos(t)} -2 $$ and now I wanted to plug in the initial conditions and this is...
  4. MechEEE

    Transfer function with initial conditions (DE)

    I have a differential equation of the form y''(t)+y'(t)+y(t)+C = 0. I think this implies that there are non-zero initial conditions. Is it possible to write a transfer function for this system? This post...
  5. Safinaz

    B How to make this integral with initial conditions

    Hello! The integral in equation (16), at the paper, is: ##I = r \int_{-\pi}^{\pi} e^{-2kr\phi} ~d\phi ## My integration is as the following : ## I = - \frac{1}{2 k} e^{-2kr\phi} ~|_{-\pi}^{\pi} + C ##, so ## I = - \frac{1}{2 k} ( e^{-2kr\pi} -e^{2kr\pi})+ C ## Now how to use the initial...
  6. VexCarido

    Runge-Kutta Projectile Approximation From Initial Conditions

    Hi everyone. I'm a new member, great to be here:) I have a few questions that I wanted to ask you guys regarding the method by which we implement the Runge-Kutta approximation of Projectile Motion if we should do it using a numerical iterative method with a Spreadsheet like Excel. I have...
  7. S

    I Laws of physics from initial conditions?

    Are there any models, theories or physicists who propose that the fundamental laws of nature come from the initial conditions? Are there any physicists who propose that the most fundamental laws of physics emerged from initial conditions at the origin of the universe? And according to this view...
  8. A

    I Non-linear ODE: initial conditions

    Say you have the set of coupled, non-linear ODEs as derived in this thread, it has two unknowns ##N(t)## and ##\theta(t)##: $$ N - mg = - m\frac{L}{2}\left(\dot{\theta}^2\cos(\theta) + \ddot{\theta}\sin(\theta)\right)$$ $$ \frac{L}{2}N\sin(\theta) = \frac{1}{12}ml^2\ddot{\theta}$$ What freedom...
  9. jisbon

    Engineering S-domain analysis with initial conditions

    I've attempted this questions after understanding the theory from my lecture notes, but my equation is looking bizarre here. Is this correct so far? Are my mesh equations correct here? Thanks.
  10. Joa

    A Finding initial conditions

    Hello, I am trying to solve a differential equation corresponding to a visco-elastic material model consisting of 5 units of springs and dashpots connected in parallel as can be seen in the image below. I am able to come-up with a single fifth order ODE, however I am struggling to find the...
  11. Quantum Alchemy

    B How constrained are the initial conditions of the universe?

    Do we know? Do we have any idea? There seems to be 2 schools of thought. 1. The initial conditions can be almost any value with universes inflating with different laws of physics. This is the multiverse and string theory 10^500 false vacua view. 2. The initial conditions are more restrained...
  12. Like Tony Stark

    Initial conditions in simple harmonic motion

    I have the formula for amplitude ##A=\sqrt (x_0^2 + \frac{\dot x_0 ^2}{\omega^2})##. But ##x_0## and ##\dot x_0## refers to the initial conditions, and the information that I'm given is not related to the initial conditions, or at least I'm not told so.
  13. C

    Engineering Advanced Circuits, Laplace Transform, Find Initial Conditions

    Vo(S) = [ N(s)Vi(s) + (- s2 + s - 2) ] / s3 + s2 + 1 ; can ignore (-s^2 + s - 2). From relevant equations: Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d3Vo(t)/dt3) + (d2Vo(t)/dt2) + Vo(t) = N(t)(dvi)/dt L[vi(t)] = t to s domain: [s3Vo(s) - s2Vo(0-) - SV'o(0-) - Vo''(0-)]Vo(s) + s2 - SVo -...
  14. Mr Davis 97

    Find which initial conditions lead to convergence

    Homework Statement Let ##b_1\in \mathbb{R}## be given and ##n=1,2,\dots## let $$b_{n+1} := \frac{1+b_n^2}{2}.$$ Define the set $$B := \{b_1\in\mathbb{R} \mid \lim_{n\to\infty}b_n \text{ converges}\}$$ Identify the set ##B##. Homework EquationsThe Attempt at a Solution I claim that ##B =...
  15. M

    MHB Romeo and Juliet initial conditions

    Hi all! I need to give a presentation on a problem in class and @countryboy helped me to figure out most of it, I only have one remaining question. Here it goes: Consider the system dr/dt = -j, dj/dt = r , where r (t) represents Romeo’s love (positive values) or hate (negative values) for...
  16. S

    A What are typical initial conditions for the Schrödinger eq?

    Hi, I am wondering if there exists some general initial conditions for solving the Schödinger eqn. for 1D free electrons ? Thanks!
  17. S

    I Can initial conditions for an ODE be given by functions instead of constants?

    Hi, I am trying to solve an ODE, however, the initial conditions are not known. From PDE examples, which are quite different, I see that some examples have initial conditions given by functions, and not by constants, i.e:: y(0) = x^2 I may have not modeled the problem correctly yet, however, I...
  18. A

    Why Set the Inner Boundary Condition to -1 in Radial Flow Equations?

    Hello gents, Q:/ what is the reason for letting the inner boundary condition = (-1) when solving the radial flow of infinite form of diffusivity equation, and i would like to know what will happened if i didn't equate it with (-1). as in the attached pic: https://i.imgur.com/AVesmHM.png
  19. A

    Supressed harmonics with certain initial conditions

    I'm currently reading class notes from an introductory waves course, written by the professor himself. I'm stuck in the Fourier analysis part, because he gives the formulas for the nth mode amplitude of a standing wave with fixed ends and then states some properties which I can't really make...
  20. karush

    MHB -17.2.53 Find the unique solution from initial conditions.

    the function y_p is a particular solution to the specified nonhomogeneous equation. Find the unique solution satisfying the equation and the given initial conditions. $y'' + 7y' = 7x$ $y_p=\frac{1}{2}x^2-\frac{1}{7}x$ $y(0)=y'(0)=0$ $r^2+7r=0$ $r=0$ $r=-7$
  21. Vincent Isoz

    Self Buckling Initial conditions

    Hi For a book I'm writing, i try to write a very very very detail proof of self-buckling. I did it almost by taking inspiration of: https://en.wikipedia.org/wiki/Self-buckling But i really really don't get how we arrive to, that when, x=0 as we obviously must have: \dfrac{\mathrm{d}^2...
  22. K

    Integral with initial conditions

    Homework Statement Homework Equations $$\frac{dy}{dx}=f(x)~\rightarrow~dy=f(x)dx~\rightarrow~y=\int f(x)dx$$ $$F=ma,~a=\frac{dv}{dt}$$ The Attempt at a Solution $$-\frac{mgR}{s^2}=mv\left( \frac{dv}{ds} \right)\rightarrow~v^2=\frac{2gR}{s}+C$$ $$v_0(s=R)=\sqrt{2gR}\rightarrow~C=2g(R-1)$$...
  23. mattyboson12

    Control systems -- input with non-zero initial conditions

    Homework Statement Deriving an s-domain equation for the following inputs a) &b) The Attempt at a Solution I understand how to derive the equation for an input with zero initial conditions (part a) but I'm not sure what to do when there are non-zero initial conditions (part b)
  24. Mr Davis 97

    Initial conditions for recurrence relation

    Homework Statement If ##a_n## counts the number of ways to climb a flight of n stairs if one can take 1, 2, or 3 steps at a time, then ##a_n = a_{n-1} + a_{n-2} + a_{n-3}##. What are the three initial conditions? Homework EquationsThe Attempt at a Solution I would say that ##a_0 = 1## since...
  25. kostoglotov

    Is there a mistake in the assignment?

    Homework Statement I'm actually a tutor, and a student of mine at uni has the following differential equation with initial conditions to solve imgur link: http://i.imgur.com/ptuymQv.gif From y(t) = c_1sin(3t) + c_2cos(3t), it is not possible to solve for the constants using the given...
  26. Yukterez

    I Schwarzschild equation of motion: initial conditions

    The Schwarzschild equation of motion, where coordinate length is differentiated by proper time is as far as I know r''(t) = -\frac{G\cdot M}{r(t)^2} + r(t)\cdot{\theta}'(t)^2 - \frac{3\cdot G\cdot M\cdot{\theta}'(t)^2}{c^2} {\theta}''(t) = -2\cdot r'(t)/r(t)\cdot{\theta}'(t) Now the question...
  27. V

    Initial Conditions Applied to a Lagrangian

    Homework Statement The scenario is a pendulum of length l and mass m2 attached to a mass of m1 which is allowed to slide along the horizontal with no friction. The support mass moves along in the X direction and the pendulum swings through the x-y plane with an angle θ with the vertical. After...
  28. M

    Ending point for initial condition theories

    I am not a scientist, but as a hobby I am summarizing different initial condition theories, specifically, eternal inflation, LGC, cyclic, and bounce theories. I need a completion time ATB where all theories produce an identical plasma. The plasma then enters the big bang process of expansion...
  29. W

    Transfer function with non zero initial conditions

    Hello guys, I'd just like to ask how can I formulate transfer function of second order differential system when I don't have zero initial conditions? the equation is = y''(t) + B/m*y'(t) + k/m*y(t) = g y(0)= -L don't care what parameters mean .. it's supposed to be solved in general Thank...
  30. ognik

    MHB Help with ODE initial conditions

    The ODE is $y'' + 4y' - 12y = 0$, I get $y = C_1e^{-6x} + C_2e^{2} $ The initial conditions are y(0) = 1, y(1)=2 - which gives me $C_1 = 1-C_2$ and $C_2 = \frac{2e^{6}-1}{e^{8}-1} $ This just looks more messy than book exercises normally are, and when I laboriously substitute back into the...
  31. T

    Solving a PDE w/ given boundary and initial conditions

    Firstly, my main question boils down to speaking about the initial conditions and boundary conditions. I was given: $$ u(0,y,t) = u(\pi,y,t) = u(x,0,t) = u(x,\pi,t) = 0 $$ but then the initial condition was: $$ u(x,y,0) = 1 $$ Aren't the initial and boundary conditions inconsistent in such...
  32. C

    Truss Free Body Diagram & initial conditions

    Homework Statement Complete a free body diagram for the truss structure with a weight attached to node 2 (this is known as 151N) Homework EquationsThe Attempt at a Solution I have attached my Free Body Diagram The diagram is to display what is know at the start of the problem before working...
  33. T

    Damping and Initial Conditions

    Hello, I have a question regarding the solution to a second order 'mass-spring-damper' system. Over the years, I have gotten familiar with the idea of system damping in the sense of under damped, over damped, and critically damped systems. However, I've began looking closer at the solution to...
  34. G

    Throw of a die: Dependance on initial conditions and chaos

    Hi, We normally use a simple symmetry argument to show that the probability of each outcome of a throw of a fair, cube-shaped die is 1/6. However, is it possible to actually model the physics of the throw and show that the probabilities are 1/6? Since this is classical physics, the outcome can...
  35. Zafa Pi

    Sensitivity to initial conditions

    Let M = {x1, x2, x3, ...} U {p} be a perfect metric space. Let f be continuous, taking M to M with f(xn) = xn+1 and f(p) = p. I would like to know if this dynamical system is necessarily sensitive to initial conditions.
  36. J

    Normal distribution starting with a uniformed distribution

    Hi comunity! I need to make a code o a normal distribution of velocities, starting whit a random secuence uniformly distributed between [0,1]. I am using FNT95, with Plato. I want to obtain a ''for'' bucle with I=1,N for the velocities. It is importan for the distribution to have sigma defined...
  37. S

    Phase Difference with Initial Conditions for SHM

    Homework Statement A mass-spring system with a natural frequency of 3.6 Hz is started in motion with an initial displacement from equilibrium of 6.1 cm and an initial velocity of 0.7 m/s. What is the value of ϕ? (Question aside: Finding the amplitude of the resulting function?) Homework...
  38. A

    Simple Harmonic Motion, Initial Displacement vs Initial Cond

    Hi everybody, I'm writing an exploration on the mathematics of simple harmonic motion and I stumbled across something I fail to understand in one of my resources (http://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx). In the example the author uses toward the end of the resource, the...
  39. B

    Determine initial conditions of the capacitor's Vc and inductor's Il at t=0

    Homework Statement Given the following circuit: [/B] Where R=C=L=1, with Vs(t) = sin(wt) and complete response Vo(t)=A*sin(wt + π/4). Homework Equations Determine the init. cond. of the capacitor voltage Vc and Inductor current Ic at t = 0. Also, find A and w.[/B]The Attempt at a Solution...
  40. E

    Material Model: initial conditions of equation?

    Hello guys, actually I'am working on a model to describe the creep-behaviour of a concrete specimen. This model is based on a 'generalized maxwell-chain'. I attached a picture, where you can see it: The acting load is the tension \sigma, while E_0 and E_1 are the stiffnesses of the...
  41. C

    Second Order ODE, With Initial Conditions

    Homework Statement y'' + 4y = t2 + 6et; y(0) = 0; y'(0) = 5 Homework Equations The Attempt at a Solution So, getting the general solution, we have r2 + 4 = 0, so r = +/- 2i So the general solution is yc = sin(2t) + cos(2t) I then used the method of undetermined coefficients to figure that...
  42. WannabeNewton

    Initial conditions for tensor perturbations

    Hi all. Say we have a background inflaton field ##\varphi## and that we've integrated the background equation for ##\varphi##, ##H(\eta)##, and ##a(\eta)## up to the number of e-folds of inflation corresponding to ##\epsilon = 1## in the slow-roll parameter. We then wish to solve for the ##k##...
  43. A

    Partial Differential Equation with initial conditions

    Hello! This is my first post to this excellent forum! I would like some help with this exercise: u_{xx} (x,y) + u_{yy} (x,y) = 0, with 0 < x < 2 \pi , 0 < y < 4 \pi u_x (0,y) = 0, \, u_x(2 \pi, y) = 0, \, 0< y < 4 \pi u(x,0) = a \cos(2x), \, u(x, 4 \pi) = a \cos^3(x), \, 0<x<2\pi...
  44. M

    A question about uniqueness of initial conditions

    Hİ. How can we sure that the initial conditions , say, for a second-order linear equation must be unique which is also the uniqueness of the solution.
  45. D

    Engineering Solving RL Circuit Differential Equation: Initial Conditions

    Homework Statement Consider the RL circuit shown in the figure. Assume that the current ##i(t)## has reached a steady state with the switch at position ##A##. At time ##t = 0##, the switch is moved from position ##A## to position ##B##. http://imgur.com/dRIOrp0 If I use the image button...
  46. Z

    Burgers Equation Question - cannot satisfy initial conditions

    Burgers Equation Question -- cannot satisfy initial conditions Homework Statement Use characteristics to solve u_t+uu_x=0 on half line x≥0 with u(x,0)=x^2 Homework Equations NA The Attempt at a Solution I think I have an issue with the initial condition. So solving via...
  47. J

    How is the initial conditions for PDE?

    Given a PDE of order 1 and another of order 2, you could show me what is, or which are, all possible initial conditions? For an ODE of order 2, for example, the initial condition is simple, is (t₀, y₀, y'₀). However, for a PDE, I think that there is various way to specify the initial condition...
  48. L

    Simple initial conditions problem

    Homework Statement http://imgur.com/Mwin7dB http://imgur.com/Mwin7dB Homework Equations The Attempt at a Solution This is a fairly simple problem. My issue is that I can't identify the second initial condition. The first one is simple. At t=0, the voltage on the capacitor is...
  49. I

    Solving 2nd Order Differential Equation with Initial Conditions

    Mod note: Reinstated problem after poster deleted it. Homework Statement Just wondering if I did this correctly: ##y''+4y'+4y=e^{x}## and initial conditions ##y(0)=0; y'(0)=1## Homework Equations The Attempt at a Solution So I found the characteristic equation to be...