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searching for Initial value problem 68 found (137 total)

Cauchy boundary condition (627 words) [view diff] no match in snippet view article find links to article

In mathematics, a Cauchy (French: [koʃi]) boundary condition augments an ordinary differential equation or a partial differential equation with conditions

James W. York (276 words) [view diff] exact match in snippet view article find links to article

Carolina State University. York used conformal geometry in the initial value problem, and introduced concepts now called the York curvature and York

Duhamel's principle (1,941 words) [view diff] exact match in snippet view article find links to article

that it is possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance

Heinz-Otto Kreiss (378 words) [view diff] exact match in snippet view article find links to article

died in Stockholm in 2015, aged 85. Kreiss did research on the initial value problem for partial differential equations, numerical treatment of partial

Beam propagation method (985 words) [view diff] no match in snippet view article find links to article

(for the waveguide axis) and they can be solved as "initial" value problem. The "initial" value problem does not involve time, rather it is for the spatial

Hadamard's method of descent (256 words) [view diff] exact match in snippet view article find links to article

wave equation, the heat equation and other versions of the Cauchy initial value problem. As Hadamard (1923) wrote: We thus have a first example of what

Carathéodory's existence theorem (1,138 words) [view diff] exact match in snippet view article find links to article

( t , t 0 , y 0 ) {\displaystyle y(t)=y(t,t_{0},y_{0})} to the initial value problem y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 . {\displaystyle

Zero stability (355 words) [view diff] exact match in snippet view article find links to article

refers to the stability of a numerical scheme applied to the simple initial value problem y ′ ( x ) = 0 {\displaystyle y'(x)=0} . A linear multistep method

Ultrahyperbolic equation (562 words) [view diff] exact match in snippet view article find links to article

and Steven Weinstein proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one hypersurface

Heat equation (9,836 words) [view diff] exact match in snippet view article find links to article

treatment. In one variable, the Green's function is a solution of the initial value problem (by Duhamel's principle, equivalent to the definition of Green's

Parabolic partial differential equation (1,176 words) [view diff] no match in snippet view article find links to article

{\displaystyle u_{t}=Lu} (note the absence of a minus sign). An initial-value problem for the backward heat equation, { u t = − Δ u on Ω × ( 0 , T

Luis Vega (mathematician) (380 words) [view diff] exact match in snippet view article

Iberoamericana (since 2011). In 2006 Vega was Invited Speaker with talk The initial value problem for nonlinear Schrödinger equations at the International Congress

Flow (mathematics) (2,679 words) [view diff] exact match in snippet view article

{\boldsymbol {x}}:\mathbb {R} \to \mathbb {R} ^{n}} the solution of the initial value problem x ˙ ( t ) = F ( x ( t ) ) , x ( 0 ) = x 0 . {\displaystyle {\dot

Green's function (5,109 words) [view diff] no match in snippet view article find links to article

{\displaystyle \delta } is Dirac's delta function; the solution of the initial-value problem L y = f {\displaystyle Ly=f} is the convolution ( G ∗ f {\displaystyle

Fourier integral operator (681 words) [view diff] exact match in snippet view article find links to article

of Fourier integral operators is the solution operator for the initial value problem for the wave operator. Indeed, consider the following problem: 1

Integral curve (816 words) [view diff] exact match in snippet view article find links to article

that α is a local solution to the ordinary differential equation/initial value problem α ( t 0 ) = p ; {\displaystyle \alpha (t_{0})=p;\,} α ′ ( t ) =

Hitoshi Ishii (430 words) [view diff] exact match in snippet view article find links to article

"偏微分方程式の初期値問題のLP可解性及び一意性" (translation: "Lp solvability and uniqueness of the initial value problem for partial differential equations"). He became an assistant professor

Caratheodory-π solution (748 words) [view diff] exact match in snippet view article find links to article

control, u = g ( x , t ) {\displaystyle u=g(x,t)} . Then, given an initial value problem, Ross partitions the time interval [ 0 , ∞ ) {\displaystyle [0,\infty

Differential inclusion (1,072 words) [view diff] exact match in snippet view article find links to article

closed, convex set for all t and x. Existence of solutions for the initial value problem d x d t ( t ) ∈ F ( t , x ( t ) ) , x ( t 0 ) = x 0 {\displaystyle

Shape dynamics (1,047 words) [view diff] exact match in snippet view article find links to article

fixed in such a way that its initial value problem and its equations of motion coincide with the initial value problem and equations of motion of the

Multiple time dimensions (1,025 words) [view diff] exact match in snippet view article find links to article

giving it the metric signature (10,2). The existence of a well-posed initial value problem for the ultrahyperbolic equation (a wave equation in more than one

Hiroshi Fujita (524 words) [view diff] exact match in snippet view article find links to article

(1962), 243–260. Hiroshi Fujita and Tosio Kato. On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964), 269–315. Hiroshi Fujita

Spacetime triangle diagram technique (2,908 words) [view diff] exact match in snippet view article find links to article

\;x_{3}=z\right\}} separation of the spatial variables result in the initial value problem for a hyperbolic PDE known as the 1D Klein–Gordon equation (KGE)

Cole–Hopf transformation (575 words) [view diff] no match in snippet view article find links to article

Cole-Hopf transformation. With the transformation, the following initial-value problem can now be solved: w t − a Δ w = 0 , w ( 0 , x ) = e − b g ( x )

Variation of parameters (3,989 words) [view diff] no match in snippet view article find links to article

obtained in this manner, for s going between 0 and t. The homogeneous initial-value problem, representing a small impulse F ( s ) d s {\displaystyle F(s)\,ds}

Straightening theorem for vector fields (413 words) [view diff] exact match in snippet view article find links to article

Let Φ ( t , p ) {\displaystyle \Phi (t,p)} be the solution of the initial value problem x ˙ = f ( x ) , x ( 0 ) = p {\displaystyle {\dot {x}}=f(x),x(0)=p}

Jean Bourgain (1,527 words) [view diff] exact match in snippet view article find links to article

group theory. He proved the uniqueness of the solutions for the initial value problem of the Korteweg–De Vries equation. He formulated what became known

John Moffat (physicist) (2,247 words) [view diff] case mismatch in snippet view article

1177–1184. (1993) "Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology," Int. Jour. Mod. Phys. D2: 351–366. (1995) "Nonsymmetric

James Serrin (440 words) [view diff] case mismatch in snippet view article find links to article

MR 0106646, S2CID 120478897, Zbl 0089.19103. Serrin, James (1963), "The initial Value problem for the Navier-Stokes equations", in Langer, Rudolph E. (ed.), Nonlinear

Richard P.A.C. Newman (432 words) [view diff] exact match in snippet view article find links to article

Online at JSTOR (accessed 17 February 2008) Discusses the Cauchy initial value problem for barotropic perfect fluid cosmological models with conformal

Mathematics of general relativity (7,038 words) [view diff] exact match in snippet view article find links to article

finite-dimensional Lie algebra. The Cauchy problem (sometimes called the initial value problem) is the attempt at finding a solution to a differential equation

Newton's law of cooling (2,814 words) [view diff] no match in snippet view article find links to article

transfer (SI unit: second − 1 {\displaystyle ^{-1}} ). Solving the initial-value problem using separation of variables gives T ( t ) = T env + ( T ( 0 )

Strang splitting (770 words) [view diff] exact match in snippet view article find links to article

coefficient matrices, then the exact solution to the associated initial value problem would be y ( t ) = e ( L 1 + L 2 ) t y 0 {\displaystyle

Steven Weinstein (philosopher) (623 words) [view diff] exact match in snippet view article

a joint paper with Walter Craig, they gave the first well-posed initial value problem for the wave equation in more than one time dimension (the ultrahyperbolic

Valentina Borok (1,082 words) [view diff] exact match in snippet view article find links to article

proved the theorem on uniqueness and well-posedness theorems for the initial value problem as well as the Cauchy problem for system of linear partial differential

Pseudo-spectral method (2,505 words) [view diff] no match in snippet view article find links to article

using fast algorithms such as the fast Fourier transform. Take the initial-value problem i ∂ ∂ t ψ ( x , t ) = [ − ∂ 2 ∂ x 2 + V ( x ) ] ψ ( x , t ) , ψ

Gigliola Staffilani (1,055 words) [view diff] exact match in snippet view article find links to article

Institutions Stanford University Brown University MIT Thesis The initial value problem for some dispersive differential equations (1995) Doctoral advisor

Continuous simulation (1,489 words) [view diff] exact match in snippet view article find links to article

problem of solving the ODEs for a given initial state is called the initial value problem. In very few cases these ODEs can be solved in a simple analytic

Truncation error (numerical integration) (1,631 words) [view diff] exact match in snippet view article

{\displaystyle p} if for any sufficiently smooth solution of the initial value problem, the local truncation error is O ( h p + 1 ) {\displaystyle O(h^{p+1})}

Mach's principle (3,121 words) [view diff] no match in snippet view article find links to article

Modern relativists see the imprints of Mach's principle in the initial-value problem. Essentially, we humans seem to wish to separate spacetime into

Adaptive step size (1,638 words) [view diff] exact match in snippet view article find links to article

their superior convergence and stability properties. Consider the initial value problem y ′ ( t ) = f ( t , y ( t ) ) , y ( a ) = y a {\displaystyle y'(t)=f(t

Kolmogorov equations (1,396 words) [view diff] no match in snippet view article find links to article

{\displaystyle i} , the Kolmogorov forward equations describe an initial-value problem for finding the probabilities of the process, given the quantities

Variable speed of light (2,598 words) [view diff] case mismatch in snippet view article find links to article

John (1993). "Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology". International Journal of Modern Physics D. 2 (3):

Lorenz system (4,388 words) [view diff] case mismatch in snippet view article find links to article

Saltzman, Barry (1962). "Finite Amplitude Free Convection as an Initial Value Problem—I". Journal of the Atmospheric Sciences. 19 (4): 329–341. Bibcode:1962JAtS

Canonical quantum gravity (3,803 words) [view diff] no match in snippet view article find links to article

James W. (1971-06-28). "Gravitational Degrees of Freedom and the Initial-Value Problem". Physical Review Letters. 26 (26): 1656–1658. Bibcode:1971PhRvL

Stochastic differential equation (5,616 words) [view diff] exact match in snippet view article find links to article

[}|Z|^{2}{\big ]}<+\infty .} Then the stochastic differential equation/initial value problem d X t = μ ( X t , t ) d t + σ ( X t , t ) d B t for t ∈ [ 0 ,

Three-wave equation (1,547 words) [view diff] no match in snippet view article find links to article

12133. Kaup, D. J. (1980). "A Method for Solving the Separable Initial-Value Problem of the Full Three-Dimensional Three-Wave Interaction". Studies in

C0-semigroup (2,590 words) [view diff] exact match in snippet view article find links to article

Cauchy problem if u(t ) ∈ D(A) for all t > 0 and it satisfies the initial value problem, a continuous function u: [0, ∞) → X is called a mild solution of

Hans Weinberger (548 words) [view diff] exact match in snippet view article find links to article

Diaz, J. B.; Weinberger, H. F. (1953). "A solution of the singular initial value problem for the Euler-Poisson-Darboux equation". Proc. Amer. Math. Soc.

Semigroup (4,673 words) [view diff] exact match in snippet view article find links to article

the above initial/boundary value problem can be interpreted as an initial value problem for an ordinary differential equation on the space X: { u ˙ ( t

Hisashi Okamoto (438 words) [view diff] exact match in snippet view article find links to article

Mathematical Sciences. Okamoto, H.; Kato, T. (1964). "On the Navier-Stokes initial value problem. I". Archive for Rational Mechanics and Analysis. 16 (4): 269–315

Hole argument (2,604 words) [view diff] exact match in snippet view article find links to article

same field equations in the original coordinate system. So the initial value problem has no unique solution in general relativity. This is also true

Boltzmann equation (4,961 words) [view diff] exact match in snippet view article find links to article

Arkeryd, Leif (1972). "On the Boltzmann equation part II: The full initial value problem". Arch. Rational Mech. Anal. 45 (1): 17–34. Bibcode:1972ArRMA..45

Camassa–Holm equation (5,872 words) [view diff] exact match in snippet view article find links to article

2006.55.2710 Camassa, Roberto (2003), "Characteristics and the initial value problem of a completely integrable shallow water equation", Discrete Contin

Integral equation (5,721 words) [view diff] exact match in snippet view article find links to article

the following section, we give an example of how to convert an initial value problem (IVP) into an integral equation. There are multiple motivations

Robin Bullough (989 words) [view diff] exact match in snippet view article find links to article

multi-soliton solutions and gone on to both introduce, and to solve the initial value problem for, the system they called the ‘Reduced Maxwell-Bloch (RMB) Equations’

Sturm–Liouville theory (4,692 words) [view diff] exact match in snippet view article find links to article

method Shooting methods proceed by guessing a value of λ, solving an initial value problem defined by the boundary conditions at one endpoint, say, a, of the

Method of averaging (4,159 words) [view diff] exact match in snippet view article find links to article

^{2}f^{[2]}(x,t,\varepsilon ).} Besides, we define the following initial value problem to be in the standard form: x ˙ = ε f 1 ( x , t ) + ε 2 f [ 2 ]

First-order partial differential equation (3,095 words) [view diff] exact match in snippet view article find links to article

shrinks with velocity c. These are light cones in space-time. The initial value problem for this equation consists in specifying a level surface S where

Lorentz group (9,740 words) [view diff] exact match in snippet view article find links to article

{\mathcal {L}}=-y\partial _{x}+x\partial _{y}.} The corresponding initial value problem (consider r = ( x , y ) {\displaystyle r=(x,y)} a function of a

Emanuele Foà (2,705 words) [view diff] case mismatch in snippet view article find links to article

ISBN 978-3-642-45916-0, MR 0108116, Zbl 0102.40503. Serrin, James (1963), "The initial Value problem for the Navier-Stokes equations", in Langer, Rudolph E. (ed.), Nonlinear

Abstract differential equation (1,944 words) [view diff] exact match in snippet view article find links to article

as a Bochner integral. The problem of finding a solution to the initial value problem d u d t = A ( t ) u + f u ( 0 ) = u 0 ∈ D ( A ) , {\displaystyle

Dario Graffi (2,956 words) [view diff] case mismatch in snippet view article find links to article

ISBN 978-3-642-45916-0, MR 0108116, Zbl 0102.40503. Serrin, James (1963), "The initial Value problem for the Navier-Stokes equations", in Langer, Rudolph E. (ed.), Nonlinear

Harold Ralph Lewis (892 words) [view diff] no match in snippet view article find links to article

time-dependent Hamiltonian systems, description of solutions of the initial-value problem for the small-signal response of collisionless plasmas, use of Hamilton's

Euler equations (fluid dynamics) (13,165 words) [view diff] exact match in snippet view article

are a subset of the conservative variables. The solution of the initial value problem in terms of characteristic variables is finally very simple. In

Theta function (14,004 words) [view diff] exact match in snippet view article find links to article

}^{\infty }\delta (x-n)} . General solutions of the spatially periodic initial value problem for the heat equation may be obtained by convolving the initial

Computational anatomy (16,865 words) [view diff] exact match in snippet view article find links to article

demonstration that the Euler–Lagrange equations have a well-posed initial value problem with unique solutions for all time, and with the first results on

Light front quantization (12,723 words) [view diff] exact match in snippet view article find links to article

equation is that the light-front is a characteristic surface for the initial value problem. This means the data on the light front is insufficient to generate