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Boundary Particle Method
In applied mathematics, the boundary particle method (BPM) is a boundary-only meshless (meshfree) collocation technique, in the sense that none of inner nodes are required in the numerical solution of nonhomogeneous partial differential equations. Numerical experiments show that the BPM has spectral convergence. Its interpolation matrix can be symmetric. History and recent developments In recent decades, the dual reciprocity method (DRM) and multiple reciprocity method (MRM) have been emerging as promising techniques to evaluate the particular solution of nonhomogeneous partial differential equations in conjunction with the boundary discretization techniques, such as boundary element method (BEM). For instance, the so-called DR-BEM and MR-BEM are popular BEM techniques in the numerical solution of nonhomogeneous problems. The DRM has become a common method to evaluate the particular solution. However, the DRM requires inner nodes to guarantee the convergence and stability. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Applied Mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the profession, professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of Mathematical analysis, applied analysis, most notably differential equations; approximation theory (broadly construed, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Plate Bending
Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load. Bending of Kirchhoff-Love plates Definitions For a thin rectangular plate of thickness H, Young's modulus E, and Poisson's ratio \nu, we can define parameters in terms of the plate deflection, w. The flexural rigidity is given by : D = \frac Moments The bending moments per unit length are given by : M_ = -D \left( \frac + \nu \frac \right) : M_ = -D \left( \nu \frac + \frac \right) The twisting moment per unit length is given by : M_ = -D \left( 1 - \nu \right) \frac Forces The shear for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Boundary Knot Method
Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the playing field, or a scoring shot where the ball is hit to or beyond that point *Boundary (sports), the sidelines of a field *Boundary (video game), ''Boundary'' (video game), a defunct 2023 multiplayer video game set in outre space Mathematics and physics *Boundary (topology), the closure minus the interior of a subset of a topological space; an edge in the topology of manifolds, as in the case of a 'manifold with boundary' *Boundary (graph theory), the vertices of edges between a subgraph and the rest of a graph *Boundary (chain complex), its abstractization in chain complexes *Boundary value problem, a differential equation together with a set of additional restraints called the boundary conditi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Method Of Fundamental Solution
Method (, methodos, from μετά/meta "in pursuit or quest of" + ὁδός/hodos "a method, system; a way or manner" of doing, saying, etc.), literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to: *Scientific method, a series of steps, or collection of methods, taken to acquire knowledge *Method (computer programming), a piece of code associated with a class or object to perform a task *Method (patent), under patent law, a protected series of steps or acts *Methodism, a Christian religious movement *Methodology, comparison or study and critique of individual methods that are used in a given discipline or field of inquiry *''Discourse on the Method'', a philosophical and mathematical treatise by René Descartes * ''Methods'' (journal), a scientific journal covering research on techniques in the experimental biological and medical sciences ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Boundary Element Method
The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, electromagnetics (where the technique is known as method of moments or abbreviated as MoM), fracture mechanics, and contact mechanics. Mathematical basis The integral equation may be regarded as an exact solution of the governing partial differential equation. The boundary element method attempts to use the given boundary conditions to fit boundary values into the integral equation, rather than values throughout the space defined by a partial differential equation. Once this is done, in the post-processing stage, the integral equation can then be used again to calculate numerically the solution directly at any desired point in the interior of the solution domain. BEM is applicable to problems for which Green's functions can be calculated. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Radial Basis Function
In mathematics a radial basis function (RBF) is a real-valued function \varphi whose value depends only on the distance between the input and some fixed point, either the origin, so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ), or some other fixed point \mathbf, called a ''center'', so that \varphi(\mathbf) = \hat\varphi(\left\, \mathbf-\mathbf\right\, ). Any function \varphi that satisfies the property \varphi(\mathbf) = \hat\varphi(\left\, \mathbf\right\, ) is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection \_k which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Low ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Meshfree Method
In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field. Motivation Numerical methods such as the finite difference method, finite-volume method, and finite element method were originally defined on meshes of data points. In such a mesh, each point has a fixed number of predefined neighbors, and this connectivity between neighbors can be used to define mathematical operators like the derivative. These operators a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Domain Decomposition
In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solution between adjacent subdomains. A coarse problem with one or few unknowns per subdomain is used to further coordinate the solution between the subdomains globally. The problems on the subdomains are independent, which makes domain decomposition methods suitable for parallel computing. Domain decomposition methods are typically used as preconditioners for Krylov space iterative methods, such as the conjugate gradient method, GMRES, and LOBPCG. In overlapping domain decomposition methods, the subdomains overlap by more than the interface. Overlapping domain decomposition methods include the Schwarz alternating method and the additive Schwarz method. Many domain decomposition methods can be written and analyzed as a special case of the abst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Trigonometric
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle measu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problem (mathematics education), word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; and they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Peter Winkler
Peter Mann Winkler is a research mathematician, author of more than 125 research papers in mathematics and patent holder in a broad range of applications, ranging from cryptography to marine navigation.Information listed oPeter Winkler's homepageat Dartmouth. His research areas include discrete mathematics, theory of computation and probability theory. He is currently a professor of mathematics and computer science at Dartmouth College. Peter Winkler studied mathematics at Harvard University and later received his PhD in 1975 from Yale University under the supervision of Angus McIntyre. He has also served as an assistant professor at Stanford, full professor and chair at Emory and as a mathematics research director at Bell Labs and Lucent Technologies. He was visiting professor at the Technische Universität Darmstadt. He has published three books on mathematical puzzles: ''Mathematical Puzzles: A connoisseur's collection'' (A K Peters, 2004, , translated to German and Russian), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |