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Katugampola Fractional Operators
In mathematics, Katugampola fractional operators are integral operators that generalize the ''Riemann–Liouville'' and the ''Hadamard'' fractional operators into a unique form.Katugampola, Udita N. (2011).On Generalized Fractional Integrals and Derivatives Ph.D. Dissertation, Southern Illinois University, Carbondale, August, 2011. The Katugampola fractional integral generalizes both the Fractional calculus#Riemann–Liouville fractional integral, Riemann–Liouville fractional integral and the Fractional calculus#Hadamard fractional integral, Hadamard fractional integral into a single form and It is also closely related to the Erdelyi–Kober operator, Erdelyi–Kober''Fractional Integrals and Derivatives: Theory and Applications'', by Samko, S.; Kilbas, A.A.; and Marichev, O. Hardcover: 1006 pages. Publisher: Taylor & Francis Books. ''Theory and Applications of Fractional Differential Equations'', by Kilbas, A. A.; Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Else ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mellin Transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of a function is :\left\(s) = \varphi(s)=\int_0^\infty x^ f(x) \, dx. The inverse transform is :\left\(x) = f(x)=\frac \int_^ x^ \varphi(s)\, ds. The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part ''c'' need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem. The transform is named after the Finnish mathematician Hjalmar Mellin, who introduced it in a paper publishe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in ... that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a function of a Complex number, complex variable s (in the complex frequency domain, also known as ''s''-domain, or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions ''f'', the Laplace transform is the integral \mathcal\(s) = \int_0^\infty f(t)e^ \, dt. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Integral
In mathematics, the Weyl integral (named after Hermann Weyl) is an operator defined, as an example of fractional calculus, on functions ''f'' on the unit circle having integral 0 and a Fourier series. In other words there is a Fourier series for ''f'' of the form : \sum_^ a_n e^ with ''a''0 = 0. Then the Weyl integral operator of order ''s'' is defined on Fourier series by : \sum_^ (in)^s a_n e^ where this is defined. Here ''s'' can take any real value, and for integer values ''k'' of ''s'' the series expansion is the expected ''k''-th derivative, if ''k'' > 0, or (−''k'')th indefinite integral normalized by integration from ''θ'' = 0. The condition ''a''0 = 0 here plays the obvious role of excluding the need to consider division by zero. The definition is due to Hermann Weyl (1917). See also *Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fig2-Katugampola
The fig is the edible fruit of ''Ficus carica'', a species of small tree in the flowering plant family Moraceae. Native to the Mediterranean and western Asia, it has been cultivated since ancient times and is now widely grown throughout the world, both for its fruit and as an ornamental plant.''The Fig: its History, Culture, and Curing'', Gustavus A. Eisen, Washington, Govt. print. off., 1901 ''Ficus carica'' is the type species of the genus ''Ficus'', containing over 800 tropical and subtropical plant species. A fig plant is a small deciduous tree or large shrub growing up to tall, with smooth white bark. Its large leaves have three to five deep lobes. Its fruit (referred to as syconium, a type of multiple fruit) is tear-shaped, long, with a green skin that may ripen toward purple or brown, and sweet soft reddish flesh containing numerous crunchy seeds. The milky sap of the green parts is an irritant to human skin. In the Northern Hemisphere, fresh figs are in season from lat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algebra gen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Operators
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: :f(t) = \int_^ (Tf) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number a+bi, is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
