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Nonlinear Partial Differential Equations
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem. The distinction between a linear and a nonlinear partial differential equation is usually made in terms of the properties of the operator that defines the PDE itself. Methods for studying nonlinear partial differential equations Existence and uniqueness of solutions A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primitive Equations
The primitive equations are a set of nonlinear partial differential equations that are used to approximate global atmosphere, atmospheric flow and are used in most Global climate model, atmospheric models. They consist of three main sets of balance equations: # A ''continuity equation'': Representing the conservation of mass. # ''Conservation of momentum'': Consisting of a form of the Navier–Stokes equations that describe hydrodynamical flow on the surface of a sphere under the assumption that vertical motion is much smaller than horizontal motion (hydrostasis) and that the fluid layer depth is small compared to the radius of the sphere # A ''Conservation of energy, thermal energy equation'': Relating the overall temperature of the system to heat sources and sinks The primitive equations may be linearized to yield Laplace's tidal equations, an eigenvalue problem from which the analytical solution to the latitudinal structure of the flow may be determined. In general, nearly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solitons
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a balanced cancellation of nonlinearity, nonlinear and dispersion relation, dispersive effects in the medium.Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency. Solitons were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Russell (engineer), John Scott Russell who observed a solitary wave in the Union Canal (Scotland), Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "John Russell (engineer)#The wave of translation, Wave of Trans ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersive Partial Differential Equation
In mathematics, a dispersive partial differential equation or dispersive PDE is a partial differential equation that is dispersive. In this context, dispersion means that waves of different wavelength propagate at different phase velocities. Examples Linear equations * Euler–Bernoulli beam equation with time-dependent loading *Airy equation *Schrödinger equation * Klein–Gordon equation Nonlinear equations *nonlinear Schrödinger equation * Korteweg–de Vries equation (or KdV equation) * Boussinesq equation (water waves) * sine–Gordon equation See also *Dispersion (optics) Dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency. Sometimes the term chromatic dispersion is used to refer to optics specifically, as opposed to wave propagation in general. A medium having this common ... * Dispersion (water waves) * Dispersionless equation References * External links *ThDispersive PDE Wiki {{mathanalysis-stub Partial differenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Scattering Transform
In mathematics, the inverse scattering transform is a method that solves the initial value problem for a Nonlinear system, nonlinear partial differential equation using mathematical methods related to scattering, wave scattering. The direct scattering transform describes how a Function (mathematics), function scatters waves or generates Bound state, bound-states. The inverse scattering transform uses wave scattering data to construct the function responsible for wave scattering. The direct and inverse scattering transforms are analogous to the direct and inverse Fourier transforms which are used to solve Linear differential equation, linear partial differential equations. Using a pair of differential operators, a 3-step algorithm may solve nonlinear system, nonlinear differential equations; the initial solution is transformed to scattering data (direct scattering transform), the scattering data evolves forward in time (time evolution), and the scattering data reconstructs the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
