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Euler–Tricomi Equation
In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow. It is named after mathematicians Leonhard Euler and Francesco Giacomo Tricomi. : u_+xu_=0. \, It is elliptic in the half plane ''x'' > 0, parabolic at ''x'' = 0 and hyperbolic in the half plane ''x'' < 0. Its characteristics are : which have the integral : where ''C'' is a constant of integration. The characteristics thus comprise two families of semicubical parabolas, with cusps on the line ''x'' = 0, the curves lying on the right hand side of the ''y''-axis. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Partial Differential Equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n-1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is : \frac = c^2 \frac The equation has the property that, if ''u'' and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), then there exists a solution for all time ''t''. The solutions of hyperbolic equations are "wave-like". If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Rel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas 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 e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burgers Equation
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948. For a given field u(x,t) and diffusion coefficient (or ''kinematic viscosity'', as in the original fluid mechanical context) \nu, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system: \frac + u \frac = \nu\frac. When the diffusion term is absent (i.e. \nu=0), Burgers' equation becomes the inviscid Burgers' equation: \frac + u \frac = 0, which is a prototype for conservation equations that can develop discontinuities (shock waves). The previous equation is the ''advective form'' of the Burgers' equation. The ''conservative form'' is found ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaplygin's Equation
In gas dynamics, Chaplygin's equation, named after Sergei Alekseevich Chaplygin (1902), is a partial differential equation useful in the study of transonic flow. It is : \frac + \frac\frac+v \frac=0. Here, c=c(v) is the speed of sound, determined by the equation of state of the fluid and conservation of energy. For polytropic gases, we have c^2/(\gamma-1) = h_0- v^2/2, where \gamma is the specific heat ratio and h_0 is the stagnation enthalpy, in which case the Chaplygin's equation reduces to : \frac + v^2\frac\frac+v \frac=0. The Bernoulli equation (see the derivation below) states that maximum velocity occurs when specific enthalpy is at the smallest value possible; one can take the specific enthalpy to be zero corresponding to absolute zero temperature as the reference value, in which case 2h_0 is the maximum attainable velocity. The particular integrals of above equation can be expressed in terms of hypergeometric functions. Derivation For two-dimensional potential flow, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semicubical Parabola
In mathematics, a cuspidal cubic or semicubical parabola is an algebraic plane curve that has an implicit equation of the form : y^2 - a^2 x^3 = 0 (with ) in some Cartesian coordinate system. Solving for leads to the ''explicit form'' : y = \pm a x^, which imply that every real point satisfies . The exponent explains the term ''semicubical parabola''. (A parabola can be described by the equation .) Solving the implicit equation for yields a second ''explicit form'' :x = \left(\frac\right)^. The parametric equation : \quad x = t^2, \quad y = a t^3 can also be deduced from the implicit equation by putting t = \frac. . The semicubical parabolas have a cuspidal singularity; hence the name of ''cuspidal cubic''. The arc length of the curve was calculated by the English mathematician William Neile and published in 1657 (see section History). Properties of semicubical parabolas Similarity Any semicubical parabola (t^2,at^3) is similar to the ''semicubical unit parabola'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Characteristics
In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface. Characteristics of first-order partial differential equation For a first-order PDE ( partial differential equation), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE. For the sake of simplicity, we confine our attention to the case of a function of two independent variabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Partial Differential Equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivative investment instruments. Definition To define the simplest kind of parabolic PDE, consider a real-valued function u(x, y) of two independent real variables, x and y. A second-order, linear, constant-coefficient PDE for u takes the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + F = 0, and this PDE is classified as being ''parabolic'' if the coefficients satisfy the condition :B^2 - AC = 0. Usually x represents one-dimensional position and y represents time, and the PDE is solved subject to prescribed initial and boundary conditions. The name "parabolic" is used because the assumption on the coefficients is the same as the condition for the analytic geometry equation A x^2 + 2B xy + C y^2 + D x + E y + F = 0 to define a planar parabola. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear relationship of voltage and current in an electrical conductor ( Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are ''nonlinear''. Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle. The word linear comes from Latin ''linearis'', "pertaining to or resembling a line". In mathematics In mathematics, a linear map or linear function ''f''(''x'') is a function that satisfies the two properties: * Additivity: . * Homogeneity of degree 1: for all α. These properties are known as the superposition principle. In this definition, ''x'' is not necessa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Partial Differential Equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form :Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\, where , , , , , , and are functions of and and where u_x=\frac, u_=\frac and similarly for u_,u_y,u_. A PDE written in this form is elliptic if :B^2-AC, applying the chain rule once gives :u_=u_\xi \xi_x+u_\eta \eta_x and u_=u_\xi \xi_y+u_\eta \eta_y, a second application gives :u_=u_ _x+u_ _x+2u_\xi_x\eta_x+u_\xi_+u_\eta_, :u_=u_ _y+u_ _y+2u_\xi_y\eta_y+u_\xi_+u_\eta_, and :u_=u_ \xi_x\xi_y+u_ \eta_x\eta_y+u_(\xi_x\eta_y+\xi_y\eta_x)+u_\xi_+u_\eta_. We can replace our PDE in x and y with an equivalent equation in \xi and \eta :au_ + 2bu_ + cu_ \text= 0,\, where :a=A^2+2B\xi_x\xi_y+C^2, :b=2A\xi_x\eta_x+2B(\xi_x\eta_y+\xi_y\eta_x) +2C\xi_y\eta_y , and :c=A^2+2B\eta_x\eta_y+C^2. To transform our PDE into the desired canonical f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francesco Giacomo Tricomi
Francesco Giacomo Tricomi (5 May 1897 – 21 November 1978) was an Italian mathematician famous for his studies on mixed type partial differential equations. He was also the author of a book on integral equations. Biography Tricomi was born in Naples. He first enrolled in the University of Bologna, where he took chemistry courses. However, Tricomi realized that he preferred physics rather than chemistry; he moved to the University of Naples in 1915. He graduated at the University of Naples in 1918 and later was assistant to Francesco Severi, first in Padua and then in Rome. Later he was professor at Turin, called by Giuseppe Peano, a position he held until his retirement in 1967. He was an Invited Speaker of the ICM in 1928 at Bologna and in 1932 in Zurich. From 1943 to 1945 and from 1948 to 1951 at the California Institute of Technology of Pasadena, he collaborated on the manual of special functions for the Bateman manuscript project, together with Arthur Erdélyi, Wilhelm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
