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Type of calculus problem
calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown
Initial_value_problem
Type of problem involving ODEs or PDEs
boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution
Boundary_value_problem
Parameter in differential equations and dynamical systems
or continuous. The problem of determining a system's evolution from initial conditions is referred to as an initial value problem. A linear matrix difference
Initial_condition
Method for solving boundary value problems
boundary value problem by reducing it to an initial value problem. It involves finding solutions to the initial value problem for different initial conditions
Shooting_method
Existence and uniqueness of solutions to initial value problems
set of sufficient (but not necessary) conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem
Picard–Lindelöf_theorem
Boundary-value problem in differential equations
or initial point. Since the parameter s {\displaystyle s} is usually time, Cauchy conditions can also be called initial value conditions or initial value
Cauchy_boundary_condition
Differential equation for the description of waves or standing wave
gauge of electromagnetism. One method to solve the initial-value problem (with the initial values as posed above) is to take advantage of a special property
Wave_equation
solution. To verify this prediction, recall the solution of the initial value problem u t t = u x x + u y y , u ( 0 , x , y ) = p ( x , y ) , u t ( 0
Constraint_counting
Partial differential equation describing the evolution of temperature in a region
given function of x and t. Initial value problem on (−∞,∞) { u t = k u x x ( x , t ) ∈ R × ( 0 , ∞ ) u ( x , 0 ) = g ( x ) Initial condition {\displaystyle
Heat_equation
Method for solving partial differential equations
possible to go from solutions of the Cauchy problem (or initial value problem) to solutions of the inhomogeneous problem. Consider, for instance, the example
Duhamel's_principle
Class of numerical methods
initial value problems. This large class of methods from numerical analysis is based on the exact integration of the linear part of the initial value
Exponential_integrator
Mathematical problem
A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant
Riemann_problem
Statement on solutions to ordinary differential equations
( 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
Carathéodory's existence theorem
Carathéodory's_existence_theorem
Procedure for solving ODEs
methods. The procedure for calculating the numerical solution to the initial value problem: y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , {\displaystyle
Heun's_method
Type of functional equation (mathematics)
However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order: f n ( x ) d n y d x n
Differential_equation
Family of implicit and explicit iterative methods
Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: d y d t = f ( t , y ) , y ( t 0 ) = y 0
Runge–Kutta_methods
Methods used to find numerical solutions of ordinary differential equations
can then be solved numerically. An Initial value problem (IVP) is a first-order differential equation plus an initial condition, where f {\displaystyle
Numerical methods for ordinary differential equations
Numerical_methods_for_ordinary_differential_equations
Theorem regarding the existence of a solution to a differential equation
theorem which guarantees the existence of solutions to certain initial value problems. Peano first published the theorem in 1886 with an incorrect proof
Peano_existence_theorem
Numerical problem-solving method
methods for solving initial value problems. This problem, in which an ordinary differential equation is given together with an initial condition, plays a
One-step_method
solution that is singular or one for which the initial value problem (also called the Cauchy problem by some authors) fails to have a unique solution
Singular_solution
Differential equation exhibiting high rate of dissipation
In computational mathematics, a stiff equation is an initial value problem u ˙ = f ( u ) , u ( 0 ) = u 0 , t ∈ [ 0 , T ] , {\displaystyle {\dot {u}}=f(u)\
Stiff_equation
Proposal in harmonic analysis
equations. In the 1970s and 1980s Jiro Takeuchi was studying the initial value problem associated with a perturbed version of the linear Schrödinger equation
Mizohata–Takeuchi_conjecture
Strong form of uniform continuity
which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is
Lipschitz_continuity
Mathematical problem solving strategy
boundary value problems. The method divides the interval over which a solution is sought into several smaller intervals, solves an initial value problem in
Direct multiple shooting method
Direct_multiple_shooting_method
Type of differential equation
{\displaystyle \psi (t)} which is the solution to the inhomogeneous initial value problem d d t ψ ( t ) = f ( ψ ( t ) , ϕ ( t − τ ) ) , {\displaystyle {\frac
Delay_differential_equation
(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
Beam_propagation_method
Problem in combinatorial optimization
The knapsack problem is the following problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine which items
Knapsack_problem
Method for solving certain nonlinear partial differential equations
transform (or nonlinear Fourier transform) is a method that solves the initial value problem for a nonlinear partial differential equation using mathematical
Inverse_scattering_transform
Existence and uniqueness theorem for certain partial differential equations
analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by Augustin Cauchy (1842), and the full
Cauchy–Kovalevskaya_theorem
Type of partial differential equations
well-posed initial value problem for the first n − 1 {\displaystyle n-1} derivatives.[citation needed] More precisely, the Cauchy problem can be locally
Hyperbolic partial differential equation
Hyperbolic_partial_differential_equation
Nondeterministic Newtonian mechanical system
\end{cases}}} Importantly these two are both solutions to the initial value problem: r ¨ = b 2 r , r ( 0 ) = 0 , r ˙ ( 0 ) = 0. {\displaystyle {\ddot
Norton's_dome
Class of problems for PDEs
are given on a hypersurface in the domain. A Cauchy problem may involve initial or boundary values. It is named after Augustin-Louis Cauchy. For a partial
Cauchy_problem
Theorem in numerical analysis
linear consistent finite difference method for a well-posed linear initial value problem, the method is convergent if and only if it is stable. The importance
Lax_equivalence_theorem
Motion of particles in a fluid
{\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
Flow_(mathematics)
Partial differential equation
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 )
Cole–Hopf_transformation
Physics problem related to laws of motion and gravity
In physics, specifically classical mechanics, the three-body problem is to take the initial positions and velocities (or momenta) of three point masses
Three-body_problem
Mathematical theorem
theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem. It is named for Thomas Hakon Grönwall
Grönwall's_inequality
Exponential representation for differential equations
Given the n × n coefficient matrix A(t), one wishes to solve the initial-value problem associated with the linear ordinary differential equation Y ′ (
Magnus_expansion
Property of differential equations describing physical phenomena
problem satisfies the following properties: It exists; It is unique; Its behavior changes continuously with the auxiliary conditions, such as initial
Well-posed_problem
Numerical method for solving ordinary differential equations
based on his earlier unpublished work. A BDF is used to solve the initial value problem y ′ = f ( t , y ) , y ( t 0 ) = y 0 . {\displaystyle y'=f(t,y),\quad
Backward differentiation formula
Backward_differentiation_formula
Class of ordinary differential equations
value of λ, solving an initial value problem defined by the boundary conditions at one endpoint, say, a, of the interval [a,b], comparing the value this
Sturm–Liouville_theory
Cosmological fine-tuning problem
cosmologists to question how the initial density came to be so closely fine-tuned to this 'special' value. The problem was first mentioned by Robert Dicke
Flatness_problem
Class of second-order linear partial differential equations
\left\{T\right\}.\end{cases}}} Similarly to a final-value problem for a parabolic PDE, an initial-value problem for a backward parabolic PDE is usually not well-posed
Parabolic partial differential equation
Parabolic_partial_differential_equation
Conjecture in general relativity
that the initial value problem for the Einstein field equations is well-posed. The conjecture then states that small perturbations of the initial conditions
Black hole stability conjecture
Black_hole_stability_conjecture
Method for solving differential equations
6}A_{1}} We can determine A0 and A1 if there are initial conditions, i.e. if we have an initial value problem. So we have A 4 = 1 4 A 2 = ( 1 4 ) ( − 1 2 )
Power series solution of differential equations
Power_series_solution_of_differential_equations
Differential equation containing derivatives with respect to only one variable
The theorem can be stated simply as follows. For the equation and initial value problem: y ′ = F ( x , y ) , y 0 = y ( x 0 ) {\displaystyle y'=F(x,y)\,
Ordinary differential equation
Ordinary_differential_equation
Dynamical system governed by Hamilton's equations
that it gives important insights into the dynamics, even if the initial value problem cannot be solved analytically. One example is the planetary movement
Hamiltonian_system
Branch of mathematics
squine functions are also uniquely determined by solving the coupled initial value problem { x ′ ( t ) = − | y ( t ) | p − 1 y ′ ( t ) = | x ( t ) | p − 1
Squigonometry
Constant expressing ambiguity from indefinite integrals
coset. In this context, solving an initial value problem is interpreted as lying in the hyperplane given by the initial conditions. Stewart, James (2008)
Constant_of_integration
Mathematical theorem
Kneser, is about the topology of the set of all solutions of an initial value problem with continuous right hand side. Consider an ordinary linear homogeneous
Kneser's theorem (differential equations)
Kneser's_theorem_(differential_equations)
Parallel algorithm from numerical analysis
algorithm from numerical analysis and used for the solution of initial value problems. It was introduced in 2001 by Lions, Maday and Turinici. Since then
Parareal
Numerical method for solving differential equations
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
Strang_splitting
Class of differential and integral operators
is the solution operator for the initial value problem for the wave operator. Indeed, consider the following problem: 1 c 2 ∂ 2 u ∂ t 2 ( t , x ) = Δ
Fourier_integral_operator
Generalisation of the exponential integral to non-commutative algebras
integral[broken anchor]. The ordered exponential is unique solution of the initial value problem: d d t OE [ a ] ( t ) = a ( t ) OE [ a ] ( t ) , OE [ a ]
Ordered_exponential
Concept that there might be more than one dimension of time
well-posed initial value problem for the ultrahyperbolic equation (a wave equation in more than one time dimension) demonstrates that initial data on a
Multiple_time_dimensions
Computer model of a physical system that continuously tracks system response
system depends on its initial state. The problem of solving the ODEs for a given initial state is called the initial value problem. In very few cases these
Continuous_simulation
Class of partial differential equations
Weinstein proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one hypersurface. And later
Ultrahyperbolic_equation
Pattern defining an infinite sequence of numbers
encounters a recurrence relation. For example, when solving the initial value problem y ′ ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , {\displaystyle
Recurrence_relation
Seven mathematical problems with a US$1 million prize for each solution
Millennium Prize Problem". OpenAI. 2026-09-08. Retrieved 2026-09-08. (initial announcement) "On the Navier–Stokes Millennium Prize Problem". OpenAI. 2026-09-08
Millennium_Prize_Problems
Term in mathematics
that α is a local solution to the ordinary differential equation/initial value problem α ( t 0 ) = p ; α ′ ( t ) = X ( α ( t ) ) . {\displaystyle {\begin{aligned}\alpha
Integral_curve
Methods of calculating definite integrals
problem of evaluating the definite integral F ( x ) = ∫ a x f ( u ) d u {\displaystyle F(x)=\int _{a}^{x}f(u)\,du} can be reduced to an initial value
Numerical_integration
Approach to finding numerical solutions of ordinary differential equations
in fact, by any other scheme for first-order systems. Given the initial value problem y ′ = y , y ( 0 ) = 1 , {\displaystyle y'=y,\quad y(0)=1,} we would
Euler_method
Belgian mathematician (1954–2018)
solutions for the initial value problem of the Korteweg–De Vries equation. He formulated what became known as the Bourgain slicing problem in high-dimensional
Jean_Bourgain
Spanish mathematician
Iberoamericana (since 2011). In 2006 Vega was Invited Speaker with talk The initial value problem for nonlinear Schrödinger equations at the International Congress
Luis_Vega_(mathematician)
Necessary condition for optimality associated with dynamic programming
optimality. The "value" of a decision problem at a certain point in time is written in terms of the payoff from some initial choices and the "value" of the remaining
Bellman_equation
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
Zero_stability
Concept in mathematics
specific values for the initial condition, one can add the plot of several solutions % solve the initial value problem symbolically % for different initial conditions
Autonomous system (mathematics)
Autonomous_system_(mathematics)
Generalized function whose value is zero everywhere except at zero
functions of x, then a convolution semigroup arises by solving the initial value problem { ∂ ∂ t η ( t , x ) = A η ( t , x ) , t > 0 lim t → 0 + η ( t ,
Dirac_delta_function
problem. The integral on the right-hand side as to be intended as a Bochner integral. The problem of finding a solution to the initial value problem d
Abstract differential equation
Abstract_differential_equation
Physical law relating heat loss to temperature difference
transfer (SI unit: second − 1 {\displaystyle ^{-1}} ). Solving the initial-value problem using separation of variables gives T ( t ) = T env + ( T ( 0 )
Newton's_law_of_cooling
Theory of gravity
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
Shape_dynamics
Classifications of Lorentzian manifolds
spacetimes the equations in general relativity can be posed as an initial value problem on a Cauchy surface. There is a hierarchy of causality conditions
Causality_conditions
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
Differential_inclusion
American physicist (1939–2023)
Carolina State University. York used conformal geometry in the initial value problem, and introduced concepts now called the York curvature and York
James_W._York
American mathematician
condition for solutions of ordinary initial value problems to be unique and to depend on a class C1 manner on the initial conditions for solutions. He died
Philip_Hartman
Mathematical concept
=x.} Let y ( t ) {\displaystyle y(t)} denote the solution to the initial value problem y ′ = y , y ( 0 ) = 1 {\displaystyle y'=y,\ y(0)=1} . Applying
Characterizations of the exponential function
Characterizations_of_the_exponential_function
Computer software bug occurring in 2038
type's maximum value is exceeded, the integer will overflow to its minimum value, which systems will interpret as in the past. The problem resembles the
Year_2038_problem
Index of articles associated with the same name
analytic partial differential equations associated with Cauchy initial value problems. Cauchy–Kowalevski–Kashiwara theorem is a wide generalization of
Uniqueness_theorem
Chaotic model of atmospheric convection
Saltzman, Barry (1962). "Finite Amplitude Free Convection as an Initial Value Problem—I". Journal of the Atmospheric Sciences. 19 (4): 329–341. Bibcode:1962JAtS
Lorenz_system
Method for solving partial differential equations
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
Hadamard's_method_of_descent
French mathematical physicist (1923–2025)
that the Einstein field equations can be expressed as a well-posed initial-value problem was listed by the journal Classical and Quantum Gravity as one of
Yvonne_Choquet-Bruhat
Procedure for solving differential equations
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}
Variation_of_parameters
Sub-area of scientific computing for solving General Relativity equations
spacetimes. In the case of dynamical spacetimes, the problem may be divided into the initial value problem and the evolution, each requiring different methods
Numerical_relativity
Class of iterative numerical methods for solving differential equations
methods for ordinary differential equations approximate solutions to initial value problems of the form y ′ = f ( t , y ) , y ( t 0 ) = y 0 . {\displaystyle
Linear_multistep_method
Differential equations involving stochastic processes
[}|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 ,
Stochastic differential equation
Stochastic_differential_equation
Functions of an angle
equation. Sine and cosine can be defined as the unique solution to the initial value problem: d d x sin x = cos x , d d x cos x = − sin x , sin
Trigonometric_functions
Millennium Prize Problem
Euclidean space, given some initial conditions and possibly an external force, has been a longstanding unsolved problem in mathematics since the early
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
states about the existence and uniqueness of the solution to an initial value problem for the first order explicit ordinary differential equation. This
Chaplygin's Theorem and Method for Solving ODE
Chaplygin's_Theorem_and_Method_for_Solving_ODE
Constraints to computational problems
time-dependent problems and multiphase flows . Transient problems require one more thing i.e., initial conditions where initial values of flow variables
Boundary conditions in fluid dynamics
Boundary_conditions_in_fluid_dynamics
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}
Straightening theorem for vector fields
Straightening_theorem_for_vector_fields
Equations characterizing continuous-time Markov processes
{\displaystyle i} , the Kolmogorov forward equations describe an initial-value problem for finding the probabilities of the process, given the quantities
Kolmogorov_equations
Equation in machine learning
{h} _{\text{in}}} of the neural ODE is obtained by solving the initial value problem d h ( t ) d t = f θ ( h ( t ) , t ) , h ( 0 ) = h in , {\displaystyle
Neural_differential_equation
Algorithm in numerical analysis
Runge-Kutta Fehlberg method with phase-lag of order infinity for initial-value problems with oscillating solution". Computers & Mathematics with Applications
Runge–Kutta–Fehlberg_method
Non-mainstream theory in physics
John (1993). "Superluminary Universe: A Possible Solution to the Initial Value Problem in Cosmology". International Journal of Modern Physics D. 2 (3):
Variable_speed_of_light
Probability puzzle
The Monty Hall problem is a brain teaser, in the form of a probability puzzle, based nominally on the American television game show Let's Make a Deal
Monty_Hall_problem
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
Caratheodory-π_solution
Topics referred to by the same term
Department of State Initial value problem is an ordinary differential equation together with a specified value, called the initial condition Intravenous
IVP
Celestial orbit whose trajectory is a conic section in the orbital plane
{\displaystyle B={\sqrt[{3}]{A+{\sqrt {A^{2}+1}}}}} This is the "initial value problem" for the differential equation (1) which is a first order equation
Kepler_orbit
Japanese mathematician
Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964), 269–315. Hiroshi Fujita. On the blowing up of solutions of the Cauchy problem for
Hiroshi_Fujita
Types of wavelets
kernel in the solution of a certain initial value problem of the Laplace operator. This is the initial value problem: Given any s ( x ) {\displaystyle s(x)}
Poisson_wavelet
travel, tourism, insurance
INITIAL VALUE-PROBLEM
INITIAL VALUE-PROBLEM
Boy/Male
Australian, Finnish, Swedish
Value; Worth; Benefit
Girl/Female
Arabic, Indian, Muslim, Parsi, Sindhi
Value; Price; Worth
Girl/Female
Indian
The initial reality
Boy/Male
Hindu, Indian
The Sprout; Initial
Girl/Female
Muslim/Islamic
Value Worth
Boy/Male
Australian, Finnish
Rule
Boy/Male
Anglo, British, English, Finnish, Swedish
Valley; Usually with a Stream; From the Glen
Boy/Male
Arabic
Value
Girl/Female
American, British, English
Of High Value
Boy/Male
Muslim
Value, Price
Girl/Female
Arabic, Muslim
Superiority; Attribute; Value
Boy/Male
Hindu, Indian
Value
Boy/Male
Indian
Value, Price
Girl/Female
Arabic
Value; Price
Boy/Male
Gujarati, Hindu, Indian
Value; Inside Trueness
Boy/Male
Arabic, Hindu, Indian, Marathi, Muslim
Powerful; Don; Value
Surname or Lastname
English
English : topographic name for someone who lived in a valley, Middle English vale (Old French val, from Latin vallis). The surname is now also common in Ireland, where it has been Gaelicized as de Bhál.Galician and Aragonese : topographic name from val ‘valley’, or habitational name from any of the places named with this word.
Girl/Female
Tamil
The initial reality
Girl/Female
American, British, English, Italian
Of High Value
Boy/Male
Arabic, Muslim
Destiny; Dignity; Value
INITIAL VALUE-PROBLEM
INITIAL VALUE-PROBLEM
INITIAL VALUE-PROBLEM
INITIAL VALUE-PROBLEM
INITIAL VALUE-PROBLEM
INITIAL VALUE-PROBLEM
INITIAL VALUE-PROBLEM
a.
Placed at the beginning; standing at the head, as of a list or series; as, the initial letters of a name.
n.
Precise signification; import; as, the value of a word; the value of a legal instrument
imp. & p. p.
of Value
n.
One who values; an appraiser.
p. pr. & vb. n.
of Initial
v. t.
To put an initial to; to mark with an initial of initials.
a.
Highly regarded; esteemed; prized; as, a valued contributor; a valued friend.
n.
Value.
v. t.
To estimate the value, or worth, of; to rate at a certain price; to appraise; to reckon with respect to number, power, importance, etc.
n.
The relative length or duration of a tone or note, answering to quantity in prosody; thus, a quarter note [/] has the value of two eighth notes [/].
v. t.
To rate highly; to have in high esteem; to hold in respect and estimation; to appreciate; to prize; as, to value one for his works or his virtues.
a.
Of or pertaining to the beginning; marking the commencement; incipient; commencing; as, the initial symptoms of a disease.
v. t.
To be worth; to be equal to in value.
v. t.
To raise to estimation; to cause to have value, either real or apparent; to enhance in value.
v. i.
Unsettled; unfixed; undetermined; indefinite; ambiguous; as, a vague idea; a vague proposition.
adv.
In an initial or incipient manner or degree; at the beginning.
imp. & p. p.
of Initial
v. i.
Proceeding from no known authority; unauthenticated; uncertain; flying; as, a vague report.
a.
Not prized or valued; being without value.
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