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List Of Nonlinear Partial Differential Equations
See also Nonlinear partial differential equation 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 mathem ..., List of partial differential equation topics and List of nonlinear ordinary differential equations. A–F : G–K : L–Q : R–Z, α–ω : References {{Reflist Partial differential equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Partial Differential Equation
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 sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camassa–Holm Equation
In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation : u_t + 2\kappa u_x - u_ + 3 u u_x = 2 u_x u_ + u u_. \, The equation was introduced by Roberto Camassa and Darryl Holm as a bi-Hamiltonian model for waves in shallow water, and in this context the parameter ''κ'' is positive and the solitary wave solutions are smooth solitons. In the special case that ''κ'' is equal to zero, the Camassa–Holm equation has peakon solutions: solitons with a sharp peak, so with a discontinuity at the peak in the wave slope. Relation to waves in shallow water The Camassa–Holm equation can be written as the system of equations: : \begin u_t + u u_x + p_x &= 0, \\ p - p_ &= 2 \kappa u + u^2 + \frac \left( u_x \right)^2, \end with ''p'' the (dimensionless) pressure or surface elevation. This shows that the Camassa–Holm equation is a model for shallow water waves with non-hydrostatic pressure and a water la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersive Long Wave Equation
Dispersive may refer to: *Dispersive partial differential equation, a partial differential equation where waves of different wavelength propagate at different phase velocities * Dispersive phase from Biological dispersal *Dispersive medium, a medium in which waves of different frequencies travel at different velocities *Dispersive adhesion, adhesion which attributes attractive forces between two materials to intermolecular interactions between molecules *Dispersive mass transfer, the spreading of mass from highly concentrated areas to less concentrated areas *Dispersive body waves, an aspect of seismic theory *Dispersive prism, an optical prism * Dispersive hypothesis, a DNA replication predictive hypothesis *Dispersive fading, in wireless communication signals *Dispersive line *Dispersive power See also * Dispersal (other) Dispersal may refer to: * Biological dispersal, the movement of organisms from their birth site to their breeding site, or from one breeding site to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degasperis–Procesi Equation
In mathematical physics, the Degasperis–Procesi equation : \displaystyle u_t - u_ + 2\kappa u_x + 4u u_x = 3 u_x u_ + u u_ is one of only two Exactly solvable model, exactly solvable equations in the following family of third-Order (differential equation), order, non-linear, dispersive PDEs: :\displaystyle u_t - u_ + 2\kappa u_x + (b+1)u u_x = b u_x u_ + u u_, where \kappa and ''b'' are real parameters (''b''=3 for the Degasperis–Procesi equation). It was discovered by Degasperis and Procesi in a search for Integrable system, integrable equations similar in form to the Camassa–Holm equation, which is the other integrable equation in this family (corresponding to ''b''=2); that those two equations are the only integrable cases has been verified using a variety of different integrability tests. Although discovered solely because of its mathematical properties, the Degasperis–Procesi equation (with \kappa > 0) has later been found to play a similar role in water wave theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Davey–Stewartson Equation
In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by to describe the evolution of a three-dimensional wave-packet on water of finite depth. It is a system of partial differential equations for a complex ( wave-amplitude) field u\, and a real ( mean-flow) field \phi\,: :i u_t + c_0 u_ + u_ = c_1 , u, ^2 u + c_2 u \phi_x,\, :\phi_ + c_3 \phi_ = ( , u, ^2 )_x.\, The DSE is an example of a soliton equation in 2+1 dimensions. The corresponding Lax representation for it is given in . In 1+1 dimensions the DSE reduces to the nonlinear Schrödinger equation :i u_t + u_ + 2k , u, ^2 u =0.\, Itself, the DSE is the particular reduction of the Zakharov–Schulman system. On the other hand, the equivalent counterpart of the DSE is the Ishimori equation. The DSE is the result of a multiple-scale analysis of modulated nonlinear surface gravity waves, propagating over a horizontal sea bed. See also * Nonlinear systems * Ishimori equation The Ishimori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calabi Conjecture
In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswald Veblen Prize in Geometry, Oswald Veblen Prize in part for his proof. His work, principally an analysis of an elliptic partial differential equation known as the Monge–Ampère equation, complex Monge–Ampère equation, was an influential early result in the field of geometric analysis. More precisely, Calabi's conjecture asserts the resolution of the prescribed Ricci curvature problem within the setting of Kähler metrics on closed manifold, closed complex manifolds. According to Chern–Weil theory, the Ricci form of any such metric is a differential form, closed differential 2-form which represents the first Chern class. Calabi conjectured that for any such differential form , there is exactly one Kähler metric in each Kähler geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Monge–Ampère Equation
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * ''Complex'' (album), by Montaigne, 2019, and its title track * ''Complex'' (EP), by Rifle Sport, 1985 * "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or more associated polypeptide chains * Spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustion does not always result in fire, because a flame is only visible when substances undergoing combustion vaporize, but when it does, a flame is a characteristic indicator of the reaction. While the activation energy must be overcome to initiate combustion (e.g., using a lit match to light a fire), the heat from a flame may provide enough energy to make the reaction self-sustaining. Combustion is often a complicated sequence of elementary radical reactions. Solid fuels, such as wood and coal, first undergo endothermic pyrolysis to produce gaseous fuels whose combustion then supplies the heat required to produce more of them. Combustion is often hot enough that incandescent light in the form of either glowing or a flame is prod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clarke's Equation
In combustion, Clarke's equation is a third-order nonlinear partial differential equation, first derived by John Frederick Clarke in 1978.Clarke, J. F. (1982). "Non-steady Gas Dynamic Effects in the Induction Domain Behind a Strong Shock Wave", College of Aeronautics report. 8229, Cranfield Inst. of Tech. https://repository.tudelft.nl/view/aereports/uuid%3A9c064b5f-97b4-4527-a97e-a805d5e1abd7 The equation describes the thermal explosion process, including both effects of constant-volume and constant-pressure processes, as well as the effects of adiabatic and isothermal sound speeds. The equation reads as :(\theta_t-\gamma e^)_=(\theta_t-e^\theta)_ where \theta is the non-dimensional temperature perturbation and \gamma is the specific heat ratio. The term \theta_t-e^\theta describes the explosion at constant pressure and the term \theta_t-\gamma e^\theta describes the explosion at constant volume. Similarly, the term (\ )_-(\ )_ describes the wave propagation at adiabatic sound sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clairaut Equation
In mathematical analysis, Clairaut's equation (or the Clairaut equation) is a differential equation of the form :y(x)=x\frac+f\left(\frac\right) where ''f'' is continuously differentiable. It is a particular case of the Lagrange differential equation. It is named after the French mathematician Alexis Clairaut, who introduced it in 1734. Definition To solve Clairaut's equation, one differentiates with respect to ''x'', yielding :\frac=\frac+x\frac+f'\left(\frac\right)\frac, so :\left +f'\left(\frac\right)\rightfrac = 0. Hence, either :\frac = 0 or :x+f'\left(\frac\right) = 0. In the former case, ''C'' = ''dy''/''dx'' for some constant ''C''. Substituting this into the Clairaut's equation, one obtains the family of straight line functions given by :y(x)=Cx+f(C),\, the so-called ''general solution'' of Clairaut's equation. The latter case, :x+f'\left(\frac\right) = 0, defines only one solution ''y''(''x''), the so-called '' singular solution'', whose graph is the en ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chafee–Infante Equation
The Chafee–Infante equation is a nonlinear partial differential equation introduced by Nathaniel Chafee and Ettore Infante.LI Zhibing Traveling wave solution of nonlinear mathematical physics equations SCIENCEP 2008(李志斌编著 《非线性数学物理方程的行波解》 科学出版社 2008) : u_t-u_+\lambda(u^3-u)=0 See also * List of nonlinear partial differential equations See also Nonlinear partial differential equation 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 gravitatio ... References #Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press # Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997 #Inna Shingareva, Carlos Lizárraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple Springer. #Eryk Infeld and George Rowlands, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
