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69-411: MADNESS ( Multiresolution Adaptive Numerical Environment for Scientific Simulation ) is a high-level software environment for the solution of integral and differential equations in many dimensions using adaptive and fast harmonic analysis methods with guaranteed precision based on multiresolution analysis and separated representations . There are three main components to MADNESS. At the lowest level

138-822: A closed bounded region in R d {\displaystyle \mathbb {R} ^{d}} with piecewise smooth boundary. The Fredholm-Volterrra Integral Operator T : C ( I × Ω ) → C ( I × Ω ) {\displaystyle {\mathcal {T}}:C(I\times \Omega )\to C(I\times \Omega )} is defined as: ( T u ) ( t , x ) := ∫ 0 t ∫ Ω K ( t , s , x , ξ ) G ( u ( s , ξ ) ) d ξ d s . {\displaystyle ({\mathcal {T}}u)(t,x):=\int _{0}^{t}\int _{\Omega }K(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds.} In

207-555: A constant factor), shifted (advanced or retarded in time) and "squeezed" or "stretched" (increasing or decreasing the frequency). The sines and cosines in the Fourier series are an example of an orthonormal basis . As an example of an application of integral transforms, consider the Laplace transform . This is a technique that maps differential or integro-differential equations in the "time" domain into polynomial equations in what

276-1016: A linear Volterra integral equation of the first kind, given by the equation: ( V y ) ( t ) = g ( t ) {\displaystyle ({\mathcal {V}}y)(t)=g(t)} can be described by the following uniqueness and existence theorem. Recall that the Volterra integral operator V : C ( I ) → C ( I ) {\displaystyle {\mathcal {V}}:C(I)\to C(I)} , can be defined as follows: ( V ϕ ) ( t ) := ∫ t 0 t K ( t , s ) ϕ ( s ) d s {\displaystyle ({\mathcal {V}}\phi )(t):=\int _{t_{0}}^{t}K(t,s)\,\phi (s)\,ds} where t ∈ I = [ t 0 , T ] {\displaystyle t\in I=[t_{0},T]} and K(t,s)

345-725: A linear Volterra integral equation of the second kind for an unknown function y ( t ) {\displaystyle y(t)} and a given continuous function g ( t ) {\displaystyle g(t)} on the interval I {\displaystyle I} where t ∈ I {\displaystyle t\in I} : y ( t ) = g ( t ) + ( V y ) ( t ) . {\displaystyle y(t)=g(t)+({\mathcal {V}}y)(t).} Volterra-Fredholm : In higher dimensions, integral equations such as Fredholm-Volterra integral equations (VFIE) exist. A VFIE has

414-458: A linear homogeneous Fredholm equation of the second type. In general, K ( x , y ) can be a distribution , rather than a function in the strict sense. If the distribution K has support only at the point x = y , then the integral equation reduces to a differential eigenfunction equation . In general, Volterra and Fredholm integral equations can arise from a single differential equation, depending on which sort of conditions are applied at

483-487: A note on naming convention: i) u(x) is called the unknown function, ii) f(x) is called a known function, iii) K(x,t) is a function of two variables and often called the Kernel function, and iv) λ is an unknown factor or parameter, which plays the same role as the eigenvalue in linear algebra . Nonlinear : An integral equation is nonlinear if the unknown function u(x) or any of its integrals appear nonlinear in

552-433: A quadrature rule Then we have a system with n equations and n variables. By solving it we get the value of the n variables Certain homogeneous linear integral equations can be viewed as the continuum limit of eigenvalue equations . Using index notation , an eigenvalue equation can be written as where M = [ M i,j ] is a matrix, v is one of its eigenvectors, and λ is the associated eigenvalue. Taking

621-399: A singular integral equation in which the kernel becomes unbounded is: x 2 = ∫ 0 x 1 x − t u ( t ) d t . {\displaystyle x^{2}=\int _{0}^{x}{\frac {1}{\sqrt {x-t}}}\,u(t)\,dt.} This equation is a special form of the more general weakly singular Volterra integral equation of

690-400: A time-domain solution. In this example, polynomials in the complex frequency domain (typically occurring in the denominator) correspond to power series in the time domain, while axial shifts in the complex frequency domain correspond to damping by decaying exponentials in the time domain. The Laplace transform finds wide application in physics and particularly in electrical engineering, where

759-648: A unique solution u ∈ C ( Ω ) {\displaystyle u\in C(\Omega )} given by: u ( t , x ) = g ( t , x ) + ∫ 0 x ∫ 0 y R ( x , ξ , y , η ) g ( ξ , η ) d η d ξ {\displaystyle u(t,x)=g(t,x)+\int _{0}^{x}\int _{0}^{y}R(x,\xi ,y,\eta )\,g(\xi ,\eta )\,d\eta \,d\xi } where R {\displaystyle R}

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828-482: Is a linear operator , since the integral is a linear operator, and in fact if the kernel is allowed to be a generalized function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem ). The general theory of such integral equations is known as Fredholm theory . In this theory, the kernel is understood to be a compact operator acting on

897-448: Is a petascale parallel programming environment that aims to increases programmer productivity and code performance/scalability while maintaining backward compatibility with current programming tools such as the message-passing interface and Global Arrays . The numerical capabilities built upon the parallel tools provide a high-level environment for composing and solving numerical problems in many (1-6+) dimensions. Finally, built upon

966-425: Is a Volterra integral equation of the form: u ( x ) = f ( x ) + ∫ α ( x ) β ( x ) K ( x , t ) ⋅ u ( t ) d t {\displaystyle u(x)=f(x)+\int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u(t)dt} where K(x,t) is called the kernel and equal to 2t , and f(x)=1 . It

1035-563: Is called an integral equation of the second kind if the unknown function also appears outside the integral. Third kind : An integral equation is called an integral equation of the third kind if it is a linear Integral equation of the following form: g ( t ) u ( t ) + λ ∫ a b K ( t , x ) u ( x ) d x = f ( t ) {\displaystyle g(t)u(t)+\lambda \int _{a}^{b}K(t,x)u(x)\,dx=f(t)} where g(t) vanishes at least once in

1104-1191: Is called the Resolvent Kernel and is given by the limit of the Neumann series for the Kernel K {\displaystyle K} and solves the resolvent equations: R ( t , s , x , ξ ) = K ( t , s , x , ξ ) + ∫ 0 t ∫ Ω K ( t , v , x , z ) R ( v , s , z , ξ ) d z d v = K ( t , s , x , ξ ) + ∫ 0 t ∫ Ω R ( t , v , x , z ) K ( v , s , z , ξ ) d z d v {\displaystyle R(t,s,x,\xi )=K(t,s,x,\xi )+\int _{0}^{t}\int _{\Omega }K(t,v,x,z)R(v,s,z,\xi )\,dz\,dv=K(t,s,x,\xi )+\int _{0}^{t}\int _{\Omega }R(t,v,x,z)K(v,s,z,\xi )\,dz\,dv} A special type of Volterra equation which

1173-1176: Is called the kernel and must be continuous on the interval D := { ( t , s ) : 0 ≤ s ≤ t ≤ T ≤ ∞ } {\displaystyle D:=\{(t,s):0\leq s\leq t\leq T\leq \infty \}} . Theorem  —  Assume that K {\displaystyle K} satisfies K ∈ C ( D ) , ∂ K / ∂ t ∈ C ( D ) {\displaystyle K\in C(D),\,\partial K/\partial t\in C(D)} and | K ( t , t ) | ≥ k 0 > 0 {\displaystyle \vert K(t,t)\vert \geq k_{0}>0} for some t ∈ I . {\displaystyle t\in I.} Then for any g ∈ C 1 ( I ) {\displaystyle g\in C^{1}(I)} with g ( 0 ) = 0 {\displaystyle g(0)=0}

1242-583: Is called the kernel and must be continuous on the interval D := { ( t , s ) : 0 ≤ s ≤ t ≤ T ≤ ∞ } {\displaystyle D:=\{(t,s):0\leq s\leq t\leq T\leq \infty \}} . Hence, the Volterra integral equation of the first kind may be written as: ( V y ) ( t ) = g ( t ) {\displaystyle ({\mathcal {V}}y)(t)=g(t)} with g ( 0 ) = 0 {\displaystyle g(0)=0} . In addition,

1311-466: Is defined as: ( T u ) ( t , x ) := ∫ 0 t ∫ Ω K ( t , s , x , ξ ) G ( u ( s , ξ ) ) d ξ d s . {\displaystyle ({\mathcal {T}}u)(t,x):=\int _{0}^{t}\int _{\Omega }K(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds.} Note that while throughout this article,

1380-456: Is identically zero. Inhomogeneous : An integral equation is called inhomogeneous if the known function f {\displaystyle f} is nonzero. Regular : An integral equation is called regular if the integrals used are all proper integrals. Singular or weakly singular : An integral equation is called singular or weakly singular if the integral is an improper integral. This could be either because at least one of

1449-401: Is often referred to as the propagator for a given system. This (physics) kernel is the kernel of the integral transform. However, for each quantum system, there is a different kernel. In the limits of integration for the inverse transform, c is a constant which depends on the nature of the transform function. For example, for the one and two-sided Laplace transform, c must be greater than

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1518-531: Is one of the codes running on the Intel MIC architecture but no performance data has been published yet. MADNESS' chemistry capability includes Hartree–Fock and density functional theory in chemistry (including analytic derivatives, response properties and time-dependent density functional theory with asymptotically corrected potentials ) as well as nuclear density functional theory and Hartree–Fock – Bogoliubov theory. MADNESS and BigDFT are

1587-622: Is one that is unchanged when the two variables are permuted; it is a kernel function K {\displaystyle K} such that K ( t , u ) = K ( u , t ) {\displaystyle K(t,u)=K(u,t)} . In the theory of integral equations, symmetric kernels correspond to self-adjoint operators . There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in their original representations. An integral transform "maps" an equation from its original "domain" into another domain, in which manipulating and solving

1656-399: Is specified by a choice of the function K {\displaystyle K} of two variables , that is called the kernel or nucleus of the transform. Some kernels have an associated inverse kernel K − 1 ( u , t ) {\displaystyle K^{-1}(u,t)} which (roughly speaking) yields an inverse transform: A symmetric kernel

1725-433: Is termed the "complex frequency" domain . (Complex frequency is similar to actual, physical frequency but rather more general. Specifically, the imaginary component ω of the complex frequency s = − σ + iω corresponds to the usual concept of frequency, viz. , the rate at which a sinusoid cycles, whereas the real component σ of the complex frequency corresponds to the degree of "damping", i.e. an exponential decrease of

1794-473: Is the primary computational kernel in MADNESS; thus, an efficient implement on modern CPUs is an ongoing research effort. . Adapting the irregular computation in MADNESS to heterogeneous platforms is nontrivial due to the size of the kernel, which is too small to be offloaded via compiler directives (e.g. OpenACC ), but has been demonstrated for CPU – GPU systems . Intel has publicly stated that MADNESS

1863-428: Is the resolvent kernel of K . As defined above, a VFIE has the form: u ( t , x ) = g ( t , x ) + ( T u ) ( t , x ) {\displaystyle u(t,x)=g(t,x)+({\mathcal {T}}u)(t,x)} with x ∈ Ω {\displaystyle x\in \Omega } and Ω {\displaystyle \Omega } being

1932-456: Is used in various applications is defined as follows: y ( t ) = g ( t ) + ( V α y ) ( t ) {\displaystyle y(t)=g(t)+(V_{\alpha }y)(t)} where t ∈ I = [ t 0 , T ] {\displaystyle t\in I=[t_{0},T]} , the function g(t) is continuous on the interval I {\displaystyle I} , and

2001-530: Is useful not only in quantum chemistry and nuclear physics, but also the modeling of partial differential equations . MADNESS was recognized by the R&;D 100 Awards in 2011. It is an important code to Department of Energy supercomputing sites and is being used by both the leadership computing facilities at Argonne National Laboratory and Oak Ridge National Laboratory to evaluate the stability and performance of their latest supercomputers. It has users around

2070-405: Is worth noting that integral equations often do not have an analytical solution, and must be solved numerically. An example of this is evaluating the electric-field integral equation (EFIE) or magnetic-field integral equation (MFIE) over an arbitrarily shaped object in an electromagnetic scattering problem. One method to solve numerically requires discretizing variables and replacing integral by

2139-414: The characteristic equations that describe the behavior of an electric circuit in the complex frequency domain correspond to linear combinations of exponentially scaled and time-shifted damped sinusoids in the time domain. Other integral transforms find special applicability within other scientific and mathematical disciplines. Another usage example is the kernel in the path integral : This states that

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2208-700: The Cauchy principal value. An Integro-differential equation, as the name suggests, combines differential and integral operators into one equation. There are many version including the Volterra integro-differential equation and delay type equations as defined below. For example, using the Volterra operator as defined above, the Volterra integro-differential equation may be written as: y ′ ( t ) = f ( t , y ( t ) ) + ( V α y ) ( t ) {\displaystyle y'(t)=f(t,y(t))+(V_{\alpha }y)(t)} For delay problems, we can define

2277-517: The Fourier transform and the Laplace transform of u(x) , respectively, with both being Fredholm equations of the first kind with kernel K ( x , t ) = e − i λ x {\displaystyle K(x,t)=e^{-i\lambda x}} and K ( x , t ) = e − λ x {\displaystyle K(x,t)=e^{-\lambda x}} , respectively. Another example of

2346-594: The Hammerstein equation and different versions of the Hammerstein equation can be found in the Hammerstein section below. First kind : An integral equation is called an integral equation of the first kind if the unknown function appears only under the integral sign. An example would be: f ( x ) = ∫ a b K ( x , t ) u ( t ) d t {\displaystyle f(x)=\int _{a}^{b}K(x,t)\,u(t)\,dt} . Second kind : An integral equation

2415-632: The Volterra integral operator ( V α t ) {\displaystyle (V_{\alpha }t)} is given by: ( V α t ) ( t ) := ∫ t 0 t ( t − s ) − α ⋅ k ( t , s , y ( s ) ) d s {\displaystyle (V_{\alpha }t)(t):=\int _{t_{0}}^{t}(t-s)^{-\alpha }\cdot k(t,s,y(s))\,ds} with ( 0 ≤ α < 1 ) {\displaystyle (0\leq \alpha <1)} . In

2484-411: The above Fredholm equation of the second kind may be written compactly as: y ( t ) = g ( t ) + λ ( F y ) ( t ) . {\displaystyle y(t)=g(t)+\lambda ({\mathcal {F}}y)(t).} Volterra : An integral equation is called a Volterra integral equation if at least one of the limits of integration is a variable. Hence,

2553-401: The amplitude.) The equation cast in terms of complex frequency is readily solved in the complex frequency domain (roots of the polynomial equations in the complex frequency domain correspond to eigenvalues in the time domain), leading to a "solution" formulated in the frequency domain. Employing the inverse transform , i.e. , the inverse procedure of the original Laplace transform, one obtains

2622-452: The boundary of the domain of its solution. Kernel (integral operator) In mathematics , an integral transform is a type of transform that 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

2691-425: The bounds of the integral are usually written as intervals, this need not be the case. In general, integral equations don't always need to be defined over an interval [ a , b ] = I {\displaystyle [a,b]=I} , but could also be defined over a curve or surface. Homogeneous : An integral equation is called homogeneous if the known function f {\displaystyle f}

2760-400: The case where the Kernel K may be written as K ( t , s , x , ξ ) = k ( t − s ) H ( x , ξ ) {\displaystyle K(t,s,x,\xi )=k(t-s)H(x,\xi )} , K is called the positive memory kernel. With this in mind, we can now introduce the following theorem: Theorem  —  If

2829-670: The consideration of the linearity of the equation or the homogeneity of the equation. These comments are made concrete through the following definitions and examples: Linear : An integral equation is linear if the unknown function u(x) and its integrals appear linearly in the equation. Hence, an example of a linear equation would be: u ( x ) = f ( x ) + λ ∫ α ( x ) β ( x ) K ( x , t ) ⋅ u ( t ) d t {\displaystyle u(x)=f(x)+\lambda \int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u(t)dt} As

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2898-409: The continuum limit, i.e., replacing the discrete indices i and j with continuous variables x and y , yields where the sum over j has been replaced by an integral over y and the matrix M and the vector v have been replaced by the kernel K ( x , y ) and the eigenfunction φ ( y ) . (The limits on the integral are fixed, analogously to the limits on the sum over j .) This gives

2967-674: The delay integral operator ( W θ , α y ) {\displaystyle ({\mathcal {W}}_{\theta ,\alpha }y)} as: ( W θ , α y ) ( t ) := ∫ θ ( t ) t ( t − s ) − α ⋅ k 2 ( t , s , y ( s ) , y ′ ( s ) ) d s {\displaystyle ({\mathcal {W}}_{\theta ,\alpha }y)(t):=\int _{\theta (t)}^{t}(t-s)^{-\alpha }\cdot k_{2}(t,s,y(s),y'(s))\,ds} where

3036-413: The delay integro-differential equation may be expressed as: y ′ ( t ) = f ( t , y ( t ) , y ( θ ( t ) ) ) + ( W θ , α y ) ( t ) . {\displaystyle y'(t)=f(t,y(t),y(\theta (t)))+({\mathcal {W}}_{\theta ,\alpha }y)(t).} The solution to

3105-406: The equation may be much easier than in the original domain. The solution can then be mapped back to the original domain with the inverse of the integral transform. There are many applications of probability that rely on integral transforms, such as "pricing kernel" or stochastic discount factor , or the smoothing of data recovered from robust statistics; see kernel (statistics) . The precursor of

3174-799: The equation. Hence, examples of nonlinear equations would be the equation above if we replaced u(t) with u 2 ( x ) , cos ⁡ ( u ( x ) ) , or  e u ( x ) {\displaystyle u^{2}(x),\,\,\cos(u(x)),\,{\text{or }}\,e^{u(x)}} , such as: u ( x ) = f ( x ) + ∫ α ( x ) β ( x ) K ( x , t ) ⋅ u 2 ( t ) d t {\displaystyle u(x)=f(x)+\int _{\alpha (x)}^{\beta (x)}K(x,t)\cdot u^{2}(t)dt} Certain kinds of nonlinear integral equations have specific names. A selection of such equations are: More information on

3243-614: The equation: u ′ ( t ) = 2 t u ( t ) , x ≥ 0 {\displaystyle u'(t)=2tu(t),\,\,\,\,\,\,\,x\geq 0} and the initial condition: u ( 0 ) = 1 {\displaystyle u(0)=1} If we integrate both sides of the equation, we get: ∫ 0 x u ′ ( t ) d t = ∫ 0 x 2 t u ( t ) d t {\displaystyle \int _{0}^{x}u'(t)dt=\int _{0}^{x}2tu(t)dt} and by

3312-412: The first kind, called Abel's integral equation: g ( x ) = ∫ a x f ( y ) x − y d y {\displaystyle g(x)=\int _{a}^{x}{\frac {f(y)}{\sqrt {x-y}}}\,dy} Strongly singular : An integral equation is called strongly singular if the integral is defined by a special regularisation, for example, by

3381-910: The following conditions: Then the VFIE has a unique solution u ∈ C ( I × Ω ) {\displaystyle u\in C(I\times \Omega )} given by u ( t , x ) = g ( t , x ) + ∫ 0 t ∫ Ω R ( t , s , x , ξ ) G ( u ( s , ξ ) ) d ξ d s {\displaystyle u(t,x)=g(t,x)+\int _{0}^{t}\int _{\Omega }R(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds} where R ∈ C ( D × Ω 2 ) {\displaystyle R\in C(D\times \Omega ^{2})}

3450-463: The following section, we give an example of how to convert an initial value problem (IVP) into an integral equation. There are multiple motivations for doing so, among them being that integral equations can often be more readily solvable and are more suitable for proving existence and uniqueness theorems. The following example was provided by Wazwaz on pages 1 and 2 in his book. We examine the IVP given by

3519-542: The following two examples are Fredholm equations: Note that we can express integral equations such as those above also using integral operator notation. For example, we can define the Fredholm integral operator as: ( F y ) ( t ) := ∫ t 0 T K ( t , s ) y ( s ) d s . {\displaystyle ({\mathcal {F}}y)(t):=\int _{t_{0}}^{T}K(t,s)\,y(s)\,ds.} Hence,

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3588-471: The following uniqueness and existence theorem. Theorem  —  Let K ∈ C ( D ) {\displaystyle K\in C(D)} and let R {\displaystyle R} denote the resolvent Kernel associated with K {\displaystyle K} . Then, for any g ∈ C ( I ) {\displaystyle g\in C(I)} ,

3657-752: The form: u ( t , x ) = g ( t , x ) + ( T u ) ( t , x ) {\displaystyle u(t,x)=g(t,x)+({\mathcal {T}}u)(t,x)} with x ∈ Ω {\displaystyle x\in \Omega } and Ω {\displaystyle \Omega } being a closed bounded region in R d {\displaystyle \mathbb {R} ^{d}} with piecewise smooth boundary. The Fredholm-Volterra Integral Operator T : C ( I × Ω ) → C ( I × Ω ) {\displaystyle {\mathcal {T}}:C(I\times \Omega )\to C(I\times \Omega )}

3726-505: The fundamental theorem of calculus, we obtain: u ( x ) − u ( 0 ) = ∫ 0 x 2 t u ( t ) d t {\displaystyle u(x)-u(0)=\int _{0}^{x}2tu(t)dt} Rearranging the equation above, we get the integral equation: u ( x ) = 1 + ∫ 0 x 2 t u ( t ) d t {\displaystyle u(x)=1+\int _{0}^{x}2tu(t)dt} which

3795-419: The integral equation above has a unique solution in y ∈ C ( I ) {\displaystyle y\in C(I)} . The solution to a linear Volterra integral equation of the second kind, given by the equation: y ( t ) = g ( t ) + ( V y ) ( t ) {\displaystyle y(t)=g(t)+({\mathcal {V}}y)(t)} can be described by

3864-817: The integral is taken over a domain varying with the variable of integration. Examples of Volterra equations would be: As with Fredholm equations, we can again adopt operator notation. Thus, we can define the linear Volterra integral operator V : C ( I ) → C ( I ) {\displaystyle {\mathcal {V}}:C(I)\to C(I)} , as follows: ( V ϕ ) ( t ) := ∫ t 0 t K ( t , s ) ϕ ( s ) d s {\displaystyle ({\mathcal {V}}\phi )(t):=\int _{t_{0}}^{t}K(t,s)\,\phi (s)\,ds} where t ∈ I = [ t 0 , T ] {\displaystyle t\in I=[t_{0},T]} and K(t,s)

3933-401: The interval [a,b] or where g(t) vanishes at a finite number of points in (a,b) . Fredholm : An integral equation is called a Fredholm integral equation if both of the limits of integration in all integrals are fixed and constant. An example would be that the integral is taken over a fixed subset of R n {\displaystyle \mathbb {R} ^{n}} . Hence,

4002-425: The largest real part of the zeroes of the transform function. Note that there are alternative notations and conventions for the Fourier transform. Here integral transforms are defined for functions on the real numbers, but they can be defined more generally for functions on a group. Although the properties of integral transforms vary widely, they have some properties in common. For example, every integral transform

4071-747: The limits of integration is infinite or the kernel becomes unbounded, meaning infinite, on at least one point in the interval or domain over which is being integrated. Examples include: F ( λ ) = ∫ − ∞ ∞ e − i λ x u ( x ) d x {\displaystyle F(\lambda )=\int _{-\infty }^{\infty }e^{-i\lambda x}u(x)\,dx} L [ u ( x ) ] = ∫ 0 ∞ e − λ x u ( x ) d x {\displaystyle L[u(x)]=\int _{0}^{\infty }e^{-\lambda x}u(x)\,dx} These two integral equations are

4140-634: The linear VFIE given by: u ( t , x ) = g ( t , x ) + ∫ 0 t ∫ Ω K ( t , s , x , ξ ) G ( u ( s , ξ ) ) d ξ d s {\displaystyle u(t,x)=g(t,x)+\int _{0}^{t}\int _{\Omega }K(t,s,x,\xi )\,G(u(s,\xi ))\,d\xi \,ds} with ( t , x ) ∈ I × Ω {\displaystyle (t,x)\in I\times \Omega } satisfies

4209-626: The numerical tools are new applications with initial focus upon chemistry, , atomic and molecular physics, material science, and nuclear structure. It is open-source , has an object-oriented design, and is designed to be a parallel processing program for computers with up to millions of cores running already on the Cray XT5 at Oak Ridge National Laboratory and the IBM Blue Gene at Argonne National Laboratory . The small matrix multiplication (relative to large, BLAS -optimized matrices)

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4278-483: The original function space using the inverse transform . An integral transform is any transform T {\displaystyle T} of the following form: The input of this transform is a function f {\displaystyle f} , and the output is another function T f {\displaystyle Tf} . An integral transform is a particular kind of mathematical operator . There are numerous useful integral transforms. Each

4347-415: The point ( x ′ , t ′ ) {\displaystyle (x',t')} multiplied by the amplitude to go from x ′ {\displaystyle x'} to x {\displaystyle x} [ i.e. K ( x , t ; x ′ , t ′ ) {\displaystyle K(x,t;x',t')} ] . It

4416-1311: The second kind can be expressed as follows: u ( t , x ) = g ( t , x ) + ∫ 0 x ∫ 0 y K ( x , ξ , y , η ) u ( ξ , η ) d η d ξ {\displaystyle u(t,x)=g(t,x)+\int _{0}^{x}\int _{0}^{y}K(x,\xi ,y,\eta )\,u(\xi ,\eta )\,d\eta \,d\xi } where ( x , y ) ∈ Ω := [ 0 , X ] × [ 0 , Y ] {\displaystyle (x,y)\in \Omega :=[0,X]\times [0,Y]} , g ∈ C ( Ω ) {\displaystyle g\in C(\Omega )} , K ∈ C ( D 2 ) {\displaystyle K\in C(D_{2})} and D 2 := { ( x , ξ , y , η ) : 0 ≤ ξ ≤ x ≤ X , 0 ≤ η ≤ y ≤ Y } {\displaystyle D_{2}:=\{(x,\xi ,y,\eta ):0\leq \xi \leq x\leq X,0\leq \eta \leq y\leq Y\}} . This integral equation has

4485-446: The second-kind Volterra integral equation has a unique solution y ∈ C ( I ) {\displaystyle y\in C(I)} and this solution is given by: y ( t ) = g ( t ) + ∫ 0 t R ( t , s ) g ( s ) d s {\displaystyle y(t)=g(t)+\int _{0}^{t}R(t,s)\,g(s)\,ds} . A Volterra Integral equation of

4554-462: The total amplitude ψ ( x , t ) {\displaystyle \psi (x,t)} to arrive at ( x , t ) {\displaystyle (x,t)} is the sum (the integral) over all possible values x ′ {\displaystyle x'} of the total amplitude ψ ( x ′ , t ′ ) {\displaystyle \psi (x',t')} to arrive at

4623-462: The transforms were the Fourier series to express functions in finite intervals. Later the Fourier transform was developed to remove the requirement of finite intervals. Using the Fourier series, just about any practical function of time (the voltage across the terminals of an electronic device for example) can be represented as a sum of sines and cosines , each suitably scaled (multiplied by

4692-442: The two most widely known codes that perform DFT and TDDFT using wavelets . Many-body wavefunctions requiring six-dimensional spatial representations are also implemented (e.g. MP2). The parallel runtime inside of MADNESS has been used to implement a wide variety of features, including graph optimization . From a mathematical perspective, MADNESS emphasizes rigorous numerical precision without loss of computational performance . This

4761-938: The world, including the United States and Japan . MADNESS has been a workhorse code for computational chemistry in the DOE INCITE program at the Oak Ridge Leadership Computing Facility and is noted as one of the important codes to run on the Cray Cascade architecture. Integral equations Various classification methods for integral equations exist. A few standard classifications include distinctions between linear and nonlinear; homogeneous and inhomogeneous; Fredholm and Volterra; first order, second order, and third order; and singular and regular integral equations. These distinctions usually rest on some fundamental property such as

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