|
Fokas Method
The Fokas method, or unified transform, is an algorithmic procedure for analysing boundary value problems for linear partial differential equations and for an important class of nonlinear partial differential equation, nonlinear PDEs belonging to the so-called integrable systems. It is named after Greek mathematician Athanassios S. Fokas. Traditionally, linear boundary value problems are analysed using either integral transforms and infinite series, or by employing appropriate fundamental solutions. Integral transforms and infinite series For example, the Dirichlet problem of the heat equation on the half-line, i.e., the problem u_0 and g_0 given, can be solved via the Sine transform, sine-transform. The analogous problem on a finite interval can be solved via an infinite series. However, the solutions obtained via integral transforms and infinite series have several disadvantages: 1. The relevant representations are not uniformly convergent at the boundaries. For example, usin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Partial Differential Equation
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] |
Generalized Dirichlet Distribution
In statistics, the generalized Dirichlet distribution (GD) is a generalization of the Dirichlet distribution with a more general covariance structure and almost twice the number of parameters. Random vectors with a GD distribution are completely neutral . The density function of p_1,\ldots,p_ is : \left \prod_^B(a_i,b_i)\right p_k^ \prod_^\left p_i^\left(\sum_^kp_j\right)^\right where we define p_k= 1- \sum_^p_i. Here B(x,y) denotes the Beta function. This reduces to the standard Dirichlet distribution if b_=a_i+b_i for 2\leqslant i\leqslant k-1 (b_0 is arbitrary). For example, if ''k=4'', then the density function of p_1,p_2,p_3 is : \left prod_^B(a_i,b_i)\right p_1^p_2^p_3^p_4^\left(p_2+p_3+p_4\right)^\left(p_3+p_4\right)^ where p_1+p_2+p_3<1 and . Connor and Mosimann define the PDF as they did for the following reason. Define random variables with |
Gauss–Hermite Quadrature
In numerical analysis, Gauss–Hermite quadrature is a form of Gaussian quadrature for approximating the value of integrals of the following kind: :\int_^ e^ f(x)\,dx. In this case :\int_^ e^ f(x)\,dx \approx \sum_^n w_i f(x_i) where ''n'' is the number of sample points used. The ''x''''i'' are the roots of the physicists' version of the Hermite polynomial ''H''''n''(''x'') (''i'' = 1,2,...,''n''), and the associated weights ''w''''i'' are given by Abramowitz, M & Stegun, I A, ''Handbook of Mathematical Functions'', 10th printing with corrections (1972), Dover, . Equation 25.4.46. :w_i = \frac . Example with change of variable Consider a function ''h(y)'', where the variable ''y'' is Normally distributed: y \sim \mathcal(\mu,\sigma^2). The expectation of ''h'' corresponds to the following integral: E(y)= \int_^ \frac \exp \left( -\frac \right) h(y) dy As this does not exactly correspond to the Hermite polynomial, we need to change variables: x = \frac \Leftrightarrow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerically
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Integral Theorem
In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if f(z) is holomorphic in a simply connected domain Ω, then for any simply closed contour C in Ω, that contour integral is zero. \int_C f(z)\,dz = 0. Statement Fundamental theorem for complex line integrals If is a holomorphic function on an open region , and \gamma is a curve in from z_0 to z_1 then, \int_f'(z) \, dz = f(z_1)-f(z_0). Also, when has a single-valued antiderivative in an open region , then the path integral \int_f'(z) \, dz is path independent for all paths in . Formulation on simply connected regions Let U \subseteq \Complex be a simply connected open set, and let f: U \to \Complex be a holomorphic function. Let \gamma: ,b\to U be a smooth closed curve. Then: \int_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan's Lemma
In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathematician Camille Jordan. Statement Consider a complex-valued, continuous function , defined on a semicircular contour :C_R = \ of positive radius lying in the upper half-plane, centered at the origin. If the function is of the form :f(z) = e^ g(z) , \quad z \in C , with a positive parameter , then Jordan's lemma states the following upper bound for the contour integral: :\left, \int_ f(z) \, dz \ \le \frac M_R \quad \text \quad M_R := \max_ \left, g \left(R e^\right) \ . with equality when vanishes everywhere, in which case both sides are identically zero. An analogous statement for a semicircular contour in the lower half-plane holds when . Remarks * If is continuous on the semicircular contour for all large and :then by Jordan's lemma \lim_ \int_ f(z)\, dz = 0. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if its Taylor series about ''x''0 converges to the function in some neighborhood for every ''x''0 in its domain. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write : f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + a_3 (x-x_0)^3 + \cdots in which the coefficients a_0, a_1, \dots are real numbers and the series is convergent to f(x) for x in a neighborhood of x_0. Alternatively, a real analytic function is an infinitely differentiable function such that the Taylor series at any point x_0 in its domain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figure The Curve
Figure may refer to: General *A shape, drawing, depiction, or geometric configuration * Figure (wood), wood appearance * Figure (music), distinguished from musical motif * Noise figure, in telecommunication * Dance figure, an elementary dance pattern *A person's figure, human physical appearance Arts *Figurine, a miniature statuette representation of a creature *Action figure, a posable jointed solid plastic character figurine * Figure painting, realistic representation, especially of the human form * Figure drawing *Model figure, a scale model of a creature Writing *figure, in writing, a type of floating block (text, table, or graphic separate from the main text) *Figure of speech, also called a rhetorical figure * Christ figure, a type of character * in typesetting, text figures and lining figures Accounting *Figure, a synonym for number * Significant figures in a decimal number Science *Figure of the Earth, the size and shape of the Earth in geodesy Sports * Figure (hor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Fourier Transform
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function f:\R \to \Complex satisfying certain conditions, and we use the convention for the Fourier transform that :(\mathcalf)(\xi):=\int_ e^ \, f(y)\,dy, then :f(x)=\int_ e^ \, (\mathcalf)(\xi)\,d\xi. In other words, the theorem says that :f(x)=\iint_ e^ \, f(y)\,dy\,d\xi. This last equation is called the Fourier integral theorem. Another way to state the theorem is that if R is the flip operator i.e. (Rf)(x) := f(-x), then :\mathcal^=\mathcalR=R\mathcal. The theorem holds if both f and its Fourier transform are absolutely integrable (in the Lebesgue sense) and f is continuous at the point x. However, even under more general ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosine Transform
In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics. Definition The Fourier sine transform of , sometimes denoted by either ^s or _s (f) , is ^s(\xi) = \int_^\infty f(t)\sin(2\pi \xi t) \,dt. If means time, then is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other. This transform is necessarily an odd function of frequency, i.e. for all : ^s(-\xi) = - ^s(\xi). The numerical factors in the Fourier transforms are defined uniquely only by their product. Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has norm of \tfrac. The Fourier cosine transform of , sometimes denoted by either ^c o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosine Transform
In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics. Definition The Fourier sine transform of , sometimes denoted by either ^s or _s (f) , is ^s(\xi) = \int_^\infty f(t)\sin(2\pi \xi t) \,dt. If means time, then is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other. This transform is necessarily an odd function of frequency, i.e. for all : ^s(-\xi) = - ^s(\xi). The numerical factors in the Fourier transforms are defined uniquely only by their product. Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has norm of \tfrac. The Fourier cosine transform of , sometimes denoted by either ^c o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
