Initial value problem
Encyclopedia
In mathematics
, in the field of differential equation
s, an initial value problem is an ordinary differential equation
together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. In physics
or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential equation is an evolution equation specifying how, given initial conditions, the system will evolve with time
.
with 
where 
is an open set,
together with a point in the domain of ƒ
called the initial condition.
A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies
This statement subsumes problems of higher order, by interpreting y as a vector.
For derivative
s of second or higher order, new variables (elements of the vector y) are introduced.
More generally, the unknown function y can take values on infinite dimensional spaces, such as Banach space
s or spaces of distribution
s.
The Picard–Lindelöf theorem
guarantees a unique solution on some interval containing t0 if ƒ is continuous on a region containing t0 and y0 and satifies the Lipschitz condition
on the variable y.
The proof of this theorem proceeds by reformulating the problem as an equivalent integral equation
. The integral can be considered an operator which maps one function into another, such that the solution is a fixed point
of the operator. The Banach fixed point theorem
is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem.
An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. Such a construction is sometimes called "Picard's method" or "the method of successive approximations". This version is essentially a special case of the Banach fixed point theorem.
Hiroshi Okamura
obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. This condition has to do with the existence of a Lyapunov function
for the system.
In some situations, the function ƒ is not of class C1
, or even Lipschitz
, so the usual result guaranteeing the local existence of a unique solution does not apply. The Peano existence theorem
however proves that even for ƒ merely continuous, solutions are guaranteed to exist locally in time; the problem is that there is no guarantee of uniqueness. The result may be found in Coddington & Levinson (1955, Theorem 1.3) or Robinson (2001, Theorem 2.6). An even more general result is the Carathéodory existence theorem, which proves existence for some discontinuous functions ƒ.
Single exponential smoothing is equivalent to computing an exponential moving average. The smoothing parameter
is determined automatically, by minimizing the squared difference between the actual and the forecast
values. Double exponential smoothing introduces a linear trend, and so has two parameters. For estimating initial value there are several methods. like we use these two formulas;
and 
. We are trying to find a formula for 
that satisfies these two equations.
Start by noting that
, so
Now rearrange the equation so that
is on the left and 
on the right
Now integrate both sides (this introduces an unknown constant
).
Eliminate the
Let
be a new unknown constant, 
, so
Now we need to find a value for
. Use 
as given at the start and substitute 0 for 
and 19 for 
this gives the final solution of
.
Second example
The solution of
can be found to be
Indeed,
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, an initial value problem is an ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential equation is an evolution equation specifying how, given initial conditions, the system will evolve with time
Time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
.
Definition
An initial value problem is a differential equation with where is an open set,together with a point in the domain of ƒ
called the initial condition.
A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies
This statement subsumes problems of higher order, by interpreting y as a vector.
For derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
s of second or higher order, new variables (elements of the vector y) are introduced.
More generally, the unknown function y can take values on infinite dimensional spaces, such as Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s or spaces of distribution
Distribution (mathematics)
In mathematical analysis, distributions are objects that generalize functions. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative...
s.
Existence and uniqueness of solutions
For a large class of initial value problems, the existence and uniqueness of a solution can be demonstrated through the use of a calculator.The Picard–Lindelöf theorem
Picard–Lindelöf theorem
In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.The theorem is named after Charles...
guarantees a unique solution on some interval containing t0 if ƒ is continuous on a region containing t0 and y0 and satifies the Lipschitz condition
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
on the variable y.
The proof of this theorem proceeds by reformulating the problem as an equivalent integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
. The integral can be considered an operator which maps one function into another, such that the solution is a fixed point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
of the operator. The Banach fixed point theorem
Banach fixed point theorem
In mathematics, the Banach fixed-point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points...
is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem.
An older proof of the Picard–Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. Such a construction is sometimes called "Picard's method" or "the method of successive approximations". This version is essentially a special case of the Banach fixed point theorem.
Hiroshi Okamura
Hiroshi Okamura
was a Japanese mathematician who studied analysis. He was a professor at Kyoto University.He discovered the necessary and sufficient conditions on initial value problems of ordinary differential equations for the solution to be unique...
obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. This condition has to do with the existence of a Lyapunov function
Lyapunov function
In the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...
for the system.
In some situations, the function ƒ is not of class C1
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
, or even Lipschitz
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
, so the usual result guaranteeing the local existence of a unique solution does not apply. The Peano existence theorem
Peano existence theorem
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy-Peano theorem, named after Giuseppe Peano and Augustin Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value...
however proves that even for ƒ merely continuous, solutions are guaranteed to exist locally in time; the problem is that there is no guarantee of uniqueness. The result may be found in Coddington & Levinson (1955, Theorem 1.3) or Robinson (2001, Theorem 2.6). An even more general result is the Carathéodory existence theorem, which proves existence for some discontinuous functions ƒ.
Exponential Smoothing
Exponential smoothing is a general method for removing noise from a data series, or producing a short term forecast of time series data.Single exponential smoothing is equivalent to computing an exponential moving average. The smoothing parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
is determined automatically, by minimizing the squared difference between the actual and the forecast
Forecasting
Forecasting is the process of making statements about events whose actual outcomes have not yet been observed. A commonplace example might be estimation for some variable of interest at some specified future date. Prediction is a similar, but more general term...
values. Double exponential smoothing introduces a linear trend, and so has two parameters. For estimating initial value there are several methods. like we use these two formulas;
Examples
A simple example is to solve and . We are trying to find a formula for that satisfies these two equations.Start by noting that
, so
Now rearrange the equation so that
is on the left and on the right
Now integrate both sides (this introduces an unknown constant
).
Eliminate the
Let
be a new unknown constant, , so
Now we need to find a value for
. Use as given at the start and substitute 0 for and 19 for 
this gives the final solution of
.Second example
The solution of
can be found to be
Indeed,
