Cauchy problem

Differential equations
Navier–Stokes differential equations used to simulate airflow around an obstruction.
Scope Natural sciences Engineering Astronomy Physics Chemistry Biology Geology Applied mathematics Continuum mechanics Chaos theory Dynamical systems Social sciences Economics Population dynamics
Classification
Types Ordinary Partial Differential-algebraic Integro-differential Fractional Linear Non-linear By variable type Dependent and independent variables Autonomous Complex Coupled / Decoupled Exact Homogeneous / Nonhomogeneous Features Order Operator Notation
Relation to processes Difference (discrete analogue) Stochastic Stochastic partial Delay
Solution
General topics Picard–Lindelöf theorem Wronskian Phase portrait Phase space Lyapunov / Asymptotic / Exponential stability Rate of convergence Series / Integral solutions Numerical integration Dirac delta function
Solution methods Inspection Method of characteristics Euler Exponential response formula Finite difference (Crank–Nicolson) Finite element Infinite element Finite volume Galerkin Petrov–Galerkin Integrating factor Integral transforms Perturbation theory Runge–Kutta Separation of variables Undetermined coefficients Variation of parameters
People Isaac Newton Gottfried Leibniz Leonhard Euler Émile Picard Józef Maria Hoene-Wroński Ernst Lindelöf Rudolf Lipschitz Augustin-Louis Cauchy John Crank Phyllis Nicolson Carl David Tolmé Runge Martin Kutta
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A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain.^[1] A Cauchy problem can be an initial value problem or a boundary value problem (for this case see also Cauchy boundary condition). It is named after Augustin-Louis Cauchy.

Formal statement

For a partial differential equation defined on Rⁿ⁺¹ and a smooth manifold S ⊂ Rⁿ⁺¹ of dimension n (S is called the Cauchy surface), the Cauchy problem consists of finding the unknown functions [math]\displaystyle{ u_1,\dots,u_N }[/math] of the differential equation with respect to the independent variables [math]\displaystyle{ t,x_1,\dots,x_n }[/math] that satisfies^[2] [math]\displaystyle{ \begin{align}&\frac{\partial^{n_i}u_i}{\partial t^{n_i}} = F_i\left(t,x_1,\dots,x_n,u_1,\dots,u_N,\dots,\frac{\partial^k u_j}{\partial t^{k_0}\partial x_1^{k_1}\dots\partial x_n^{k_n}},\dots\right) \\ &\text{for } i,j = 1,2,\dots,N;\, k_0+k_1+\dots+k_n=k\leq n_j;\, k_0\lt n_j \end{align} }[/math] subject to the condition, for some value [math]\displaystyle{ t=t_0 }[/math],

[math]\displaystyle{ \frac{\partial^k u_i}{\partial t^k}=\phi_i^{(k)}(x_1,\dots,x_n) \quad \text{for } k=0,1,2,\dots,n_i-1 }[/math]

where [math]\displaystyle{ \phi_i^{(k)}(x_1,\dots,x_n) }[/math] are given functions defined on the surface [math]\displaystyle{ S }[/math] (collectively known as the Cauchy data of the problem). The derivative of order zero means that the function itself is specified.

Cauchy–Kowalevski theorem

The Cauchy–Kowalevski theorem states that If all the functions [math]\displaystyle{ F_i }[/math] are analytic in some neighborhood of the point [math]\displaystyle{ (t^0,x_1^0,x_2^0,\dots,\phi_{j,k_0,k_1,\dots,k_n}^0,\dots) }[/math], and if all the functions [math]\displaystyle{ \phi_j^{(k)} }[/math] are analytic in some neighborhood of the point [math]\displaystyle{ (x_1^0,x_2^0,\dots,x_n^0) }[/math], then the Cauchy problem has a unique analytic solution in some neighborhood of the point [math]\displaystyle{ (t^0,x_1^0,x_2^0,\dots,x_n^0) }[/math].

See also

• Cauchy boundary condition
• Cauchy horizon

References

External links

• Cauchy problem at MathWorld.

de:Anfangswertproblem#Partielle Differentialgleichungen

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