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There is a theorem called the Cauchy-Kowalevski theoremwhich deals with the existence of solutions to a system of mdifferential equation in n dimensions when the coefficients are analytic functions. I am guessing this is what you are asking about. A special case of this theorem was proved by Cauchy alone.

The theorem talks about the local existence of a solution.

Since this is a complicated topic, I will provide a link.

Q: Cauchy problem for first order partial differential equation?

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She did not figure out a particular equation but found the set of conditions under which solutions to a class of partial differential equations would exist. This is now known as the Cauchy-Kovalevskaya Theorem.

partial of u with respect to x = partial of v with respect to y partial of u with respect to y = -1*partial of v with respect to x

Well, cauchy-riemann differential equation is a part of complex variables and in real-life applications such as engineering, it can be used in determining the flow of fluids, such as the flow around the pipe. In fluid mechanics, the cauchy-riemann equations are decribed by two complex variables, i.e. u and v, and if these two variables satisfy the equations in an open subset of R2, then the vector field can be asserted from the two cauchy-riemann equations, ux = vy (1) uy = - vx (2) This I think can help interpreting the potential flow (Wikipedia) in two dimensions using the cauchy-riemann equations. In fluid mechanics, the potential flow can be analyzed using the cauchy-riemann equations.

She is probably best known for her work on partial differential equations. Her paper on the subject contains what is now known as the Cauchy–Kovalevskaya theorem, which gives conditions under which a certain class of those equations does have solutions.For her doctorate she also presented papers, at the University of Göttingen, on the dynamics of Saturn's rings and on elliptic integrals.

(xn) is Cauchy when abs(xn-xm) tends to 0 as m,n tend to infinity.

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Sigeru Mizohata has written: 'Lectures on Cauchy problem' -- subject(s): Cauchy problem, Differential equations, Partial, Partial Differential equations

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One thing about math is that sometimes the challenge of solving a difficult problem is more rewarding than even it's application to the "real" world. And the applications lead to other applications and new problems come up with other interesting solutions and on and on... But... The Cauchy-Euler equation comes up a lot when you try to solve differential equations (the Cauchy-Euler equation is an ordinary differential equation, but more complex partial differential equations can be decomposed to ordinary differential equations); differential equations are used extensively by engineers and scientists to describe, predict, and manipulate real-world scenarios and problems. Specifically, the Cauchy-Euler equation comes up when the solution to the problem is of the form of a power - that is the variable raised to a real power. Specific cases involving equilibrium phenomena - like heat energy through a bar or electromagnetics often rely on partial differential equations (Laplace's Equation, or the Helmholtz equation, for example), and there are cases of these which can be separated into the Cauchy-Euler equation.

Lars Garding has written: 'Cauchy's problem for hyperbolic equations' -- subject(s): Differential equations, Partial, Exponential functions, Partial Differential equations 'Applications of the theory of direct integrals of Hilbert spaces to some integral and differential operators' -- subject(s): Differential equations, Partial, Hilbert space, Partial Differential equations

She did not figure out a particular equation but found the set of conditions under which solutions to a class of partial differential equations would exist. This is now known as the Cauchy-Kovalevskaya Theorem.

partial of u with respect to x = partial of v with respect to y partial of u with respect to y = -1*partial of v with respect to x

Francois Treves is an Italian mathematician known for his research in partial differential equations and functional analysis. He has authored numerous academic papers and several books, including "Basic Linear Partial Differential Equations" and "Introduction to Pseudo-Differential and Fourier Integral Operators."

relation of cauchy riemann equation in other complex theorems

Well, cauchy-riemann differential equation is a part of complex variables and in real-life applications such as engineering, it can be used in determining the flow of fluids, such as the flow around the pipe. In fluid mechanics, the cauchy-riemann equations are decribed by two complex variables, i.e. u and v, and if these two variables satisfy the equations in an open subset of R2, then the vector field can be asserted from the two cauchy-riemann equations, ux = vy (1) uy = - vx (2) This I think can help interpreting the potential flow (Wikipedia) in two dimensions using the cauchy-riemann equations. In fluid mechanics, the potential flow can be analyzed using the cauchy-riemann equations.

Well, cauchy-riemann differential equation is a part of complex variables and in real-life applications such as engineering, it can be used in determining the flow of fluids, such as the flow around the pipe. In fluid mechanics, the cauchy-riemann equations are decribed by two complex variables, i.e. u and v, and if these two variables satisfy the equations in an open subset of R2, then the vector field can be asserted from the two cauchy-riemann equations, ux = vy (1) uy = - vx (2) This I think can help interpreting the potential flow (Wikipedia) in two dimensions using the cauchy-riemann equations. In fluid mechanics, the potential flow can be analyzed using the cauchy-riemann equations.

The Cauchy kovalevskaya theorem tells us about solutions to systems of differential equations. If we look at m equations in n dimension, with coefficient that are analytic function, we can know about the existence of solutions using this theorem.

Alberto Bressan has written: 'Well-posedness of the Cauchy problem for nxn systems of conservation laws' -- subject(s): Cauchy problem, Conservation laws (Mathematics)