Initial conditions Definition and 133 Threads

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. O

    Damped oscillator given odd initial conditions

    Homework Statement (A) The damped oscillator is described by the equation mx''+bx'+kx=0. What is the condition for critical damping expressed in terms of m,b,k. Assume this is satisfied. (B) For t<0 the mass is at rest (x=0). This mass is set in motion at t=0 by a sharp impulsive force so...
  2. mesa

    Solve a differential equation for the initial conditions,

    Homework Statement Solve y' = (y^3)(t^2) for the initial condition y(0)=0 and state in which interval in 't' this solution exists. The Attempt at a Solution First I divided both sides by y^3 and then subtracted t^2 from each as well, -t^2 + y'(y^-3) = 0 then solved, (-t^3)/3 +...
  3. S

    Initial conditions in quantum mechanics

    Hello, I'm doing an undergraduate introductory course in quantum mechanics, and I'm having a hard time understanding the interpretation. I've recently learned about the famous "Particle in a box" problem, in which we solve the time-independent schrodinger wave equation to get the energy...
  4. B

    How to minimise this function and get initial conditions

    How to minimise this function and get initial conditions . I have the answer for initial conditions. ra = min(00237 − 0000175v + 8.693f − 000159y) subjected to 124.53 ≤ v ≤ 167.03 0.025 ≤ f ≤ 0.083 6.2 ≤ y ≤ 14.8 v= 144.2 , f = 0.025, y = 9.5 How to get this?
  5. P

    2nd order nonhomogeneos differential equations with initial conditions

    Homework Statement The problem states d^2y/dt^2 +15y= cost4t + 2sin t initial conditions y(0)=y'(0)=0 Homework Equations The Attempt at a Solution All I have is this r^2+15=0 making r(+-)=√15 and making yh= C1cos√15+C2√15 the next part includes solve for...
  6. P

    2nd order nonhomogeneos differential equations with initial conditions

    I have a problem which in involves a second order differential equations with imaginary roots and I can seem to know how to finish the problem. d^2y/dt^2 +15y =cos 4t+2 sin t this is what I got so far r^2+15=0 for the homogeneous part r=+-(√15) Yh=C1cos√15+C2sin√15 now is...
  7. F

    Springel Model - Initial conditions of galaxy for simulation

    Hello, I'am studying a code (called "starscream") which allows to make initial conditions (positions and velocities) for NBody simulation. They are based on Springel and White (1999) model. I have several problems about the underlying equations used in this code. Firstly, See attachment...
  8. J

    Initial Conditions of Astrophysical Simulations in 3D

    So I've been playing with a little N-body code in 3D for gravitational problems and have made an OpenGL visualization to go along with it. I have been generating initial conditions (positions and velocities) using explicit formulas. I was wondering of anyone knew of any resources for getting...
  9. R

    Solving a 2nd Order Differential Equation with Initial Conditions

    Homework Statement \frac{d^2 y}{dx^2}\cdot\frac{dy}{dx}=x(x+1), \hspace{10pt} y(0)=1, \hspace{5pt} y'(0)=2 Homework Equations None I can think of... The Attempt at a Solution The only thing I even thought to try was turn it into the form: \frac{d^2 y}{dx^2}{dy}=x(x+1){dx}...
  10. A

    PDE: Initial Conditions Contradicting Boundary Conditions

    Suppose we have the following IBVP: PDE: u_{t}=α^{2}u_{xx} 0<x<1 0<t<∞ BCs: u(0,t)=0, u_{x}(1,t)=1 0<t<∞ IC: u(x,0)=sin(πx) 0≤x≤1 It appears as though the BCs and the IC do not match. The derivative of temperature with respect to x at position x=1 is a constant 1...
  11. U

    Solving a First Order Differential Equation with Initial Conditions.

    Homework Statement Solve the initial value problem: t(dy/dt)+8y=t^3 where t>0 and y(1)=0 Homework Equations None? The Attempt at a Solution It's a linear equation, so rearranged to dy/dt+8y/t=t^2. Took the integrating factor e^(∫8/tdt)=t^8 and multiplied through...
  12. D

    MHB Solving the Heat Equation with Initial Conditions

    I have already solved the main portions. I have $$ T(x,t) = \sum_{n = 1}^{\infty}A_n\cos\lambda_n x\exp(-\lambda_n^2t) $$ The eigenvalues are determined by $$ \tan\lambda_n = \frac{1}{\lambda_n} $$ The initial condition is $T(x,0) =1$. For the particular case of $f(x) = 1$, numerically...
  13. O

    Second ODE, initial conditions are zeros at infinity

    second ODE, initial conditions are zeros at infinity! I want to know the temperature profile of phase transition layer in the interstellar medium. For stationary solution, the dimensionless differential equation I ended up with is \frac{d^2T}{dx^2} = \frac{f(T)}{T^2} - \frac{1}{T} where f(T)...
  14. T

    State-Space Representation with Initial Conditions: Finding the Right DSP Book

    Could you advise a DSP book which explains state-space representation with initial conditions. Initial conditions part is important for me. Thanks.
  15. M

    MHB Solve Heat Equation with Initial Conditions

    Solve $\begin{aligned} & {{u}_{tt}}={{u}_{xx}},\text{ }x\in [0,1],\text{ }t>0, \\ & u(x,0)=f(x), \\ & {{u}_{t}}(x,0)=0,\text{ }u(0,t)=u(1,t)=0 \\ \end{aligned} $ where $f(x)$ is defined by $f(x)=x$ if $0\le x\le \dfrac12$ and $f(x)=1-x$ if $\dfrac12\le x\le1.$ I'm not sure how to...
  16. F

    DE with initial conditions with repeated rootsplease check work thanks

    Homework Statement 4y'' - 4y' + y = 0 y(1) = -4 y'(1) = 0 Homework Equations getting the coeff's gets messier and messier.. am i doing something wrong? The Attempt at a Solution 4y'' - 4y' + y = 0 y'' - y' - (1/4)y = 0 r_1 = r_2 = 1/2 general soln: y(t) = c_1e^(t/2) +...
  17. K

    Solution of differential with initial conditions

    I think my book is giving me the wrong answer...The problem is to find solution of following: r'(t) = t2\hat{i} + 5t\hat{j} + \hat{k} The initial condition is: r(1) = \hat{j} + 2\hat{k} My solution: r(t) = < (1/3)t3 + c1 , (5/2)t2 + c2 , t+c3 > r(1) = < 0 , 1 , 2 > r(1) = <...
  18. R

    Solving Non-Linear Diffusion Eqn w/ Eigenfunction Expansion

    I'm trying to solve a non-linear time-dependent diffusion equation to find R(x,t). To do so, I'm positing that : R(x,t)=\sum^{J}_{1}X_{i}(x)T_{i}(t) which allows me to arrive at something that looks like : dT_{i}/dt=A_{i}T_{i}(t)-B*T_{i}(t)^{2} The problem I'm having, through sheer...
  19. M

    Solving Homogenous Eq. with Initial Conditions: u(x)=\int^x_0 f(s)g(x,s)ds

    Eq u'(x)+p(x)u=f(x) with initial condition u(0)=0 It's homogenous solution is u_h=Ce^{-\int^x_0 p(s)ds} Complete solution u(x)=e^{-\int^x_0 p(s)ds}\int^x_0f(s)e^{\int^s_0 p(z)dz}ds=\int^x_0 f(s)g(x,s)ds where g(x,s)=e^{-\int^{x}_s p(\xi)d \xi } I didn't see that last...
  20. maverick_starstrider

    Classical Stat Mech with Uncertain Initial Conditions vs. Quantum

    Hi, I was wondering if someone could point me to a textbook or easy to read paper (or website) that briefly describes/proves the differences here. What I mean is if I do classical (continuous energies) statistical mechanics where my initial state is a volume (greater than or equal to h-bar)...
  21. U

    Initial conditions for BH in GR

    HI all There is smth I don't understand in building initial data for BH The equations everybody uses uses spacetime or timelike hypersurfaces but for example horizon is a null surface!
  22. T

    What are the initial conditions in this simple RCL circuit?

    So, let's assume the switch has been closed for a long time. The capacitor is charged to 12V, and the coil acts like a short-circuit. Immediately after opening the switch, is it right to say the Voltage in the coil is 12V? My notes specify something along the lines of: the conditions at time...
  23. X

    Guessing functions given initial conditions

    Is there any way to mathematically derive functions that satisfy a given set of initial conditions? I know this sounds very general, but say for: f(1)-f(0)=0 and f'(1)=1 I've resolved myself to guessing and checking. I've found a function for the opposite: e*t-exp(t) -- ([e*1 -...
  24. D

    Find s(t) Given Initial Conditions and F(s)

    This problem has been bothering me for a while now, hope you can help me. Let's say that the initial velocity of an object, with mass of m is 0 and the initial position is s0 and the force acting on the object is defined as F(s), how do i find the s(t), where t is time. If it's any help, then...
  25. C

    Mathematica Mathematica: NDSolve with InterpolatingFunction as initial conditions

    The questions arises because I want to use the solution of one PDE as initial condition to solve another. Then using NDSolve, the first solution is given by InterpolatingFunction. I tried, it takes the whole afternoon, almost kills my computer but still not giving any result. Another computer...
  26. M

    Solution satisfying initial conditions for a pde of second order

    Homework Statement I have found the general solution to a second order pde to be U(x,t) = f(3x + t) + g(-x + t) where f and g are arbitrary functions I have initial conditions U(x,0) = sin(x) Du/dt (x,0) = cos (2x) The Attempt at a Solution I have found that U(x,0) = f(3x) +...
  27. M

    Using initial conditions in a second order PDE

    Homework Statement I have a PDE for which i have found the general solution to be u(x,y) = f1(3x + t) + f2(-x + t) where f1 and f2 are arbitrary functions. I have initial conditions u(x,0) = sin (x) and partial derivative du/dt (x,0) = cos (2x)Homework Equations u(x,y) = f1(3x + t) + f2(-x +...
  28. Phrak

    Initial Conditions in Classical Physics

    According to the usual way of applying determinism in physics:- If we know all the intitial conditions of a closed system at time t0, we can fully specify the the system at a time t1>t0. This seems natural and obvious within classical physics, but is it really true? I have never heard of a...
  29. G

    Solving a PDE with Characteristic Curves and Initial Conditions

    Homework Statement sin(y)\frac{ \partial u}{ \partial x} + \frac{ \partial u}{ \partial y} = (xcos(y)-sin^2(y))u where ln(u(x,\frac{\pi}{2})) = x^2 + x - \frac{\pi}{2} for -1 \leq x \leq 3 determine the characteristic curves in the xy plane and draw 3 of them determine the general...
  30. D

    State transition matrix to change initial conditions.

    Hey folks, I have an orbit in the circular restricted three body problem with initial conditions [x(0), 0, z(0), 0, y'(0), 0] I'm following this paper http://adsabs.harvard.edu/full/1984CeMec..32...53H on how to correct these initial conditions given the state transition matrix at a...
  31. D

    Initial conditions for a halo orbit

    Hey folks I'm looking into halo orbits and I have a question about how to find the initial conditions from the third order approximation solution... A good run through of the third order solution calculation is found in this paper...
  32. D

    Diffusion equation without initial conditions

    Hi, How can I solve the diffusion equation in one dimension: u_t=ku_{xx} ; -\infty < t < \infty , 0<x<\infty With the boundary conditions: u(0,t)= T_0 +Acos(wt) u(x\rightarrow \infty,t) \rightarrow T_0 Thanks!
  33. TheFerruccio

    Shallow Water Equations and Initial Conditions

    I have seen lots of simulations using the shallow water equations, and every one I've seen involves having some sort of initial displacement in the water, resulting in a propagation of waves through a medium of either constant or variable depth, as a function of position. However, can you...
  34. M

    Wave Equation with initial conditions, boundary condtions

    So, I do not think I did this properly, but if f(-x)=-f(x), then u(-x,0)=-u(x,0), and if g(-x)=-g(x), then ut(-x,0)=-ut(x,0). According to D`Alambert`s formula, u(x,t)=[f(x+t)+f(x-t)]/2 + 0.5∫g(s)ds (from x-t to x+t) so, u(0,t)=[f(t)+f(-t)]/2 + 0.5∫g(s)ds (from -t to t) f is odd, and so is...
  35. C

    Calculus of variations problem and differential equation initial conditions

    Calculus of variations problem. I want to make stationary the integral of (1+yy')^2 dx from 0 to 1. I know what the Euler-Lagrange differential equation turns out to be, but how do I interpret the limits of integration as initial conditions for the diff eq? also, can i use laplace transforms to...
  36. Y

    Coupled Oscillators Initial Conditions and Phase

    Hello I have a question about coupled oscillators and what initial conditions affect what constants of integration. In the book I have, A.P. French Vibrations and Waves, the guesses at solutions are chosen at random and sometimes do include a phase shift, while sometimes they dont. For...
  37. R

    Conservation laws, Noether's theorem and initial conditions

    Hello, everybody! During the whole of my undergraduate study of physics, this one thing always bothered me. It concerns the interplay of conserved quantities, symmetries, Noether's theorem and initial conditions. For a system of N degrees of freedom, governed by the usual Newton's laws...
  38. E

    Initial conditions for rlc series natural response

    Homework Statement Find v(t) across a cap. in a series rlc circuit with no driving force (initial v across cap: 24V) Homework Equations from the values of the components, \alpha > \omega_0, the circuit is overdamped, and the following equation can be used: v(t) =A_1 e^{s_1 t} + A_2...
  39. C

    Solving Problem with Derivatives as Initial Conditions

    Homework Statement I've been given equations that have derivatives as initial conditions, rather than things like u(0,t)=u(L,t)=0 Things like this: http://img444.imageshack.us/img444/5082/mathu.th.jpg Uploaded with ImageShack.us Homework Equations The Attempt at a...
  40. S

    Inflation and initial conditions

    Its been argued that inflation needs the initial conditions to be finely tuned: eg: http://en.wikipedia.org/wiki/Inflation_(cosmology)#Fine-tuning_problem does this paper resolve this problem...
  41. B

    Solve Differential eq with initial conditions

    Homework Statement Solve: \frac{\partial H}{\partial t} = -4\kappa H^{2} With initial condition: H(0) = 1/L^{2} To find: H(t) = \frac{1}{4\kappa t + L^{2}} 2. The attempt at a solution I tried using Taylor series expansion such that: H(t)\approx H(0)+t\frac{\partial...
  42. T

    What Variables are Zero in Different Thermodynamic Conditions?

    Hello Everyone, Is there a list online somewhere of which variable values are 0 and why (conceptually), based on initial conditions? Like I know that for a system at constant volume, w=0. I am looking for a list for the other effects like isothermal, adiabatic, what does pressure= if there is...
  43. T

    NDSolve Quirkiness - Updating the initial conditions after each step?

    I'm modeling a physical system described by a second-order ODE with LOTS of parameters. Using SciLab, I sucessfully implimented the model using a while loop with time. Because of the loop structure, every time I called the DE solver (ode), I updated the intial conditions for the current...
  44. M

    Nonhomogeneous wave equation with vanishing initial conditions

    Homework Statement Let u(x,t) be the solution of the following initial value problem for the nonhomogeneous wave equation, u_{x_1x_1}+u_{x_2x_2}+u_{x_3x_3}-u_{tt}=f(x_1,x_2,x_3,t) u(x,0)=0 and u_t(x,0)=0 x\in\Re^3 , t>0 Use Duhamel's Principle and Kirchoff's formula to show that...
  45. P

    What Determines the Maximum Velocity of a Particle in a Given Energy System?

    The total energy of a particle is given by: E_{tot} = 2 \dot{x}^{2} - cos(\frac{1}{2} \pi x) and I'm told that the particle passed through the point x=1 [m] with a velocity of \vec{v}=-\frac{1}{2} \hat{x} [m/sec]. I'm required to find the maximum velocity of the particle during its motion...
  46. fluidistic

    Classical mechanics, initial conditions question

    I just started CM (I had 2 classes until now) and the professor said that if you know the position and velocity of say all the particles, then you know how the system will evolve. This, I already read and knew. I've probably a common question so feel free to redirect me to any similar...
  47. E

    Heat Equation Initial Conditions

    Greetings all, I have a question in regards to my initial conditions. The problem as given is: ut=uxx with u' = 0 at x=0 and u=0 at x=L I was also given u={1 0<x<L/2, 0 L/2<x<L I understand the set up of the problem and the solving of it for the most part, however I'm having...
  48. N

    Why Is the Voltage on Both Capacitors 12 Volts in This Circuit?

    http://i48.tinypic.com/1o4ow9.jpg why in this circuit they say that in t<0 the voltage on both capacitators is 12 volts?? if both points of each capacitators is connected to the voltage source directy then 12 volts is ok. but here we have a resistor between them this resistor drops...
  49. N

    Solve First order linear differential equation, initial conditions

    Homework Statement The problem is given as follows: Solve dy/dt + y = 0.5, y(t=0)=1Homework Equations The Attempt at a Solution I separate the y terms from the t terms, which gives me dy(-y+0.5)=dt I integrate both sides to get -ln(-y+0.5)=t+C C is the constant, I combine the constants from...
  50. M

    Solving for y in an ODE with Initial Conditions

    what is the solution for y in this peculiar ODE ? A\left(y,x\right)=\frac{dy}{dx}+B(x)(1-y) with initial conditions : \frac{dy}{dx}=\left0 \ldots , y=0 \frac{dy}{dx}=\delta(x-x_{0})\ldots,y=1 moreover \int^{\infty}_{-\infty}Adx=\int^{\infty}_{-\infty}Bdx=1
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