In
multivariable calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather th ...
, an initial value problem (IVP) is an
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
together with an
initial condition
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 ...
which specifies the value of the unknown
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
at a given point in the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
. Modeling a system in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system
evolves with time given the initial conditions of the problem.
Definition
An initial value problem is a differential equation
:
with
where
is an open set of
,
together with a point in the domain of
:
called the
initial condition
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 solution to an initial value problem is a function
that is a solution to the differential equation and satisfies
:
In higher dimensions, the differential equation is replaced with a family of equations
, and
is viewed as the vector
, most commonly associated with the position in space. More generally, the unknown function
can take values on infinite dimensional spaces, such as
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s or spaces of
distributions.
Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g.
.
Existence and uniqueness of solutions
The
Picard–Lindelöf theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauc ...
guarantees a unique solution on some interval containing ''t''
0 if ''f'' is continuous on a region containing ''t''
0 and ''y''
0 and satisfies the
Lipschitz condition
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exi ...
on the variable ''y''.
The proof of this theorem proceeds by reformulating the problem as an equivalent
integral equation
In mathematics, integral equations are equations in which an unknown Function (mathematics), function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; ...
. 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
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...
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
was a Japanese mathematician who made contributions to analysis and the theory of differential equations. He was a professor at Kyoto University.''Funkcialaj Ekvacioj'', 2 (1959), Profesoro Hirosi OKAMURA, nekrologo (''E-e'')
He discovered the n ...
obtained a
necessary and sufficient condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
for the solution of an initial value problem to be unique. This condition has to do with the existence of a
Lyapunov function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s se ...
for the system.
In some situations, the function ''f'' is not of
class ''C''1, or even
Lipschitz, so the usual result guaranteeing the local existence of a unique solution does not apply. The
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 t ...
however proves that even for ''f'' 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 ''f''.
Examples
A simple example is to solve
and
. We are trying to find a formula for
that satisfies these two equations.
Rearrange the equation so that
is on the left hand side
:
Now integrate both sides with respect to
(this introduces an unknown constant
).
:
:
Eliminate the logarithm with exponentiation on both sides
:
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,
:
Notes
See also
*
Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
*
Constant of integration
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
*
Integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name
Integral curves are known by various other names, depending on the nature and interpreta ...
References
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