Showing papers on "Elementary function published in 2010"
TL;DR: A survey of more advanced topics in automatic differentiation includes an introduction to the reverse mode (the authors' implementation is forward mode) and considerations in arbitrary-order multivariable series computation.
Abstract: An introduction to both automatic differentiation and object-oriented programming can enrich a numerical analysis course that typically incorporates numerical differentiation and basic MATLAB computation. Automatic differentiation consists of exact algorithms on floating-point arguments. This implementation overloads standard elementary operators and functions in MATLAB with a derivative rule in addition to the function value; for example, $\sin u$ will also compute $(\cos u)\ast u^{\prime}$, where $u$ and $u^{\prime }$ are numerical values. These methods are mostly one-line programs that operate on a class of value-and-derivative objects, providing a simple example of object-oriented programming in MATLAB using the new (as of release 2008a) class definition structure. The resulting powerful tool computes derivative values and multivariable gradients, and is applied to Newton's method for root-finding in both single and multivariable settings. To compute higher-order derivatives of a single-variable function, another class of series objects keeps Taylor polynomial coefficients up to some order. Overloading multiplication on series objects is a combination (discrete convolution) of coefficients. This idea leads to algorithms for other operations and functions on series objects. A survey of more advanced topics in automatic differentiation includes an introduction to the reverse mode (our implementation is forward mode) and considerations in arbitrary-order multivariable series computation.
200 citations
TL;DR: Holonomic functions as discussed by the authors are a class of functions that can be characterized by sufficiently many partial differential and difference equations, both linear and with polynomial coefficients, and can be expressed as holonomic systems.
Abstract: The holonomic systems approach was proposed in the early 1990s by Doron Zeilberger. It laid a foundation for the algorithmic treatment of holonomic function identities. Frederic Chyzak later extended this framework by introducing the closely related notion of ∂-finite functions and by placing their manipulation on solid algorithmic grounds. For practical purposes it is convenient to take advantage of both concepts which is not too much of a restriction: The class of functions that are holonomic and ∂-finite contains many elementary functions (such as rational functions, algebraic functions, logarithms, exponentials, sine function, etc.) as well as a multitude of special functions (like classical orthogonal polynomials, elliptic integrals, Airy, Bessel, and Kelvin functions, etc.). In short, it is composed of functions that can be characterized by sufficiently many partial differential and difference equations, both linear and with polynomial coefficients. An important ingredient is the ability to execute closure properties algorithmically, for example addition, multiplication, and certain substitutions. But the central technique is called creative telescoping which allows to deal with summation and integration problems in a completely automatized fashion.Part of this thesis is our Mathematica package HolonomicFunctions in which the above mentioned algorithms are implemented, including more basic functionality such as noncommutative operator algebras, the computation of Grobner bases in them, and finding rational solutions of parameterized systems of linear differential or difference equations.Besides standard applications like proving special function identities, the focus of this thesis is on three advanced applications that are interesting in their own right as well as for their computational challenge. First, we contributed to translating Takayama's algorithm into a new context, in order to apply it to an until then open problem, the proof of Ira Gessel's lattice path conjecture. The computations that completed the proof were of a nontrivial size and have been performed with our software. Second, investigating basis functions in finite element methods, we were able to extend the existing algorithms in a way that allowed us to derive various relations which generated a considerable speed-up in the subsequent numerical simulations, in this case of the propagation of electromagnetic waves. The third application concerns a computer proof of the enumeration formula for totally symmetric plane partitions, also known as Stembridge's theorem. To make the underlying computations feasible we employed a new approach for finding creative telescoping operators.
181 citations
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18 Oct 2010
TL;DR: In this article, a series of functions of complex variables are presented, including Integrals Integral Transforms Orthogonal Curvilinear Systems of Coordinates Ordinary Differential Equations.
Abstract: MATHEMATICS Arithmetic and Elementary Algebra Elementary Functions Elementary Geometry Analytic Geometry Algebra Limits and Derivatives Integrals Series Functions of Complex Variables Integral Transforms Ordinary Differential Equations Partial Differential Equations Special Functions and Their Properties Probability Theory PHYSICS Physical Foundations of Mechanics Molecular Physics and Thermodynamics Electrodynamics Oscillations and Waves Optics Quantum Mechanics. Atomic Physics Quantum Theory of Crystals Elements of Nuclear Physics ELEMENTS OF APPLIED AND ENGINEERING SCIENCES Dimensions and Similarity Mechanics of Point Particles and Rigid Bodies Elements of Strength of Materials Hydrodynamics Mass and Heat Transfer Electrical Engineering Empirical and Engineering Formulas and Criteria for Their Applicability SUPPLEMENTS Integrals Integral Transforms Orthogonal Curvilinear Systems of Coordinates Ordinary Differential Equations Some Useful Electronic Mathematical Resources Index References appear at the end of each chapter.
39 citations
TL;DR: Different explicit equations expressed in terms of the elementary functions are proposed here as useful shortcuts to fit time depletion of substrate concentration directly to progress curves using commonly available nonlinear regression computer programs.
36 citations
TL;DR: Two widely used methods for computing matrix exponentials and matrix logarithms are improved by exploiting the special structure of skew-symmetric and orthogonal matrices by combining Pade approximation and scaling and squaring.
36 citations
TL;DR: In this paper, a set of 3D general solutions for thermoporoelastic media for the steady-state problem is presented, which can serve as a benchmark for various kinds of numerical codes and approximate solutions.
Abstract: This paper presents a set of 3D general solutions for thermoporoelastic media for the steady-state problem. By introducing two displacement functions, the equations governing the elastic, pressure and temperature fields are simplified. The operator theory and superposition principle are then employed to express all the physical quantities in terms of two functions, one of which satisfies a quasi–Laplace equation and the other satisfies a differential equation of the eighth order. The generalized Almansi's theorem is used to derive the displacements, pressure and temperature in terms of five quasi-harmonic functions for various cases of material characteristic roots. To show its practical significance, an infinite medium containing a penny-shaped crack subjected to mechanical, pressure and temperature loads on the crack surface is given as an example. A potential theory method is employed to solve the problem. One integro-differential equation and two integral equations are derived, which bear the same structures to those reported in literature. For a penny-shaped crack subjected to uniformly distributed loads, exact and complete solutions in terms of elementary functions are obtained, which can serve as a benchmark for various kinds of numerical codes and approximate solutions.
34 citations
TL;DR: In this article, the behavior of a radiating star when the interior expanding, shearing fluid particles are traveling in geodesic motion is studied and new classes of exact solutions in terms of elementary functions without assuming a separable form for the gravitational potentials or initially fixing the temporal evolution of the model unlike earlier treatments.
Abstract: We study the behavior of a radiating star when the interior expanding, shearing fluid particles are traveling in geodesic motion. We demonstrate that it is possible to obtain new classes of exact solutions in terms of elementary functions without assuming a separable form for the gravitational potentials or initially fixing the temporal evolution of the model unlike earlier treatments. A systematic approach enables us to write the junction condition as a Riccati equation which under particular conditions may be transformed into a separable equation. New classes of solutions are generated which allow for mixed spatial and temporal dependence in the metric functions. We regain particular models found previously from our general classes of solutions.
26 citations
TL;DR: In this article, a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains, and solutions of the Cauchy-Dirichlet and cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions.
Abstract: In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.
26 citations
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TL;DR: In this article, saddle point integrals of real and imaginary phase functions are studied and the results are generalized to the case where the phase function is analytic and non-degenerate.
Abstract: We consider saddle point integrals in d variables whose phase func- tions are neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The proofs are via contour shifting in complex d-space. This work is motivated by applications to asymptotic enumeration. arise in many areas of mathematics. There are many variations. This integral may involve one or more variables; the variables may be real or complex; the integral may be global or taken over a small neighborhood or oddly shaped set; varying degrees of smoothness may be assumed; and varying degrees of degeneracy may be allowed near the critical points of the phase function, φ. Often what is sought is a leading order estimate of I(λ) as the positive real parameter λ tends to ∞ ,o r an asymptotic series I(λ) ∼ � n cngn(λ), where {gn} is a sequence of elementary functions and the expansion is possibly nowhere convergent, but satisfies
21 citations
TL;DR: In this paper, the authors study the behavior of a radiating star when the interior expanding, shearing fluid particles are traveling in geodesic motion and obtain new classes of exact solutions in terms of elementary functions without assuming a separable form for the gravitational potentials.
Abstract: We study the behaviour of a radiating star when the interior expanding, shearing fluid particles are traveling in geodesic motion. We demonstrate that it is possible to obtain new classes of exact solutions in terms of elementary functions without assuming a separable form for the gravitational potentials or initially fixing the temporal evolution of the model unlike earlier treatments. A systematic approach enables us to write the junction condition as a Riccati equation which under particular conditions may be transformed into a separable equation. New classes of solutions are generated which allow for mixed spatial and temporal dependence in the metric functions. We regain particular models found previously from our general classes of solutions.
19 citations
TL;DR: In this article, a closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions is obtained and a trial approximate solution depending on some parameters is used, which is substituted in the tension equation.
Abstract: A closed-form approximate expression for the solution of a simple pendulum in terms of elementary functions is obtained. To do this, the exact expression for the maximum tension of the string of the pendulum is first considered and a trial approximate solution depending on some parameters is used, which is substituted in the tension equation. We obtain the parameters for the approximate by means of a term-by-term comparison of the power series expansion for the approximate maximum tension with the corresponding series for the exact one. We believe that this letter may be a suitable and fruitful exercise for teaching and better understanding nonlinear oscillations of a simple pendulum in undergraduate courses on classical mechanics.
TL;DR: Efficient methods are proposed for evaluating the sine, cosine, arc tangent, exponential and logarithmic functions in double precision without table look-ups, scattering from, or gathering into SIMD registers, or conditional branches.
Abstract: Data-parallel architectures like SIMD (Single Instruction Multiple Data) or SIMT (Single Instruction Multiple Thread) have been adopted in many recent CPU and GPU architectures. Although some SIMD and SIMT instruction sets include double-precision arithmetic and bitwise operations, there are no instructions dedicated to evaluating elementary functions like trigonometric functions in double precision. Thus, these functions have to be evaluated one by one using an FPU or using a software library. However, traditional algorithms for evaluating these elementary functions involve heavy use of conditional branches and/or table look-ups, which are not suitable for SIMD computation. In this paper, efficient methods are proposed for evaluating the sine, cosine, arc tangent, exponential and logarithmic functions in double precision without table look-ups, scattering from, or gathering into SIMD registers, or conditional branches. We implemented these methods using the Intel SSE2 instruction set to evaluate their accuracy and speed. The results showed that the average error was less than 0.67 ulp, and the maximum error was 6 ulps. The computation speed was faster than the FPUs on Intel Core 2 and Core i7 processors.
TL;DR: In this paper, a function theory in higher dimensions based on hyperbolic metric was proposed, and the advantage of this theory is that positive and negative powers of hyper complex variables are included to the theory, which is not in the monogenic case.
Abstract: The theory of monogenic functions or regular functions is based on Euclidean metric. We consider a function theory in higher dimensions based on hyperbolic metric. The advantage of this theory is that positive and negative powers of hyper complex variables are included to the theory, which is not in the monogenic case. Hence elementary functions can be defined similarly as in classical complex analysis.
TL;DR: In this paper, a Crocco-type transformation is presented, which reduces the order of the Navier-Stokes equation for the longitudinal component of the velocity, and problems concerning the nonlinear stability/instability of the solutions thus obtained are investigated.
Abstract: New classes of exact solutions of three-dimensional nonstationary Navier-Stokes equations are described. These solutions contain arbitrary functions. Many periodic solutions (both with respect to the spatial coordinate and with respect to time) and aperiodic solutions are obtained, which can be expressed in terms of elementary functions. A Crocco-type transformation is presented, which reduces the order of the equation for the longitudinal component of the velocity. Problems concerning the nonlinear stability/instability of the solutions thus obtained are investigated. It turns out that a specific feature of many solutions of the Navier-Stokes equations is their instability. It is shown that instability can take place not only for rather large Reynolds numbers but also for arbitrarily small ones (and can be independent of the velocity profile of the fluid). A general physical interpretation and classification of solutions is given.
TL;DR: The architecture is optimized by analyzing the trade-off between the size of the required memory and the precision of intermediate variables to achieve the minimum 23-bit accuracy required for single-precision floating-point representation.
Abstract: This paper presents a unified architecture for the compact implementation of several key elementary functions, including reciprocal, square root, and logarithm, in single-precision floating-point arithmetic. The proposed high-throughput design is based on uniform domain segmentation and curve fitting techniques. Numerically accurate least-squares regression is utilized to calculate the polynomial coefficients. The architecture is optimized by analyzing the trade-off between the size of the required memory and the precision of intermediate variables to achieve the minimum 23-bit accuracy required for single-precision floating-point representation. The efficiency of the proposed unified data path is demonstrated on a common field-programmable gate array.
TL;DR: The 2-D Green's function for a steady line heat source in a semi-infinite piezothermoelectric plane is presented by four newly induced harmonic functions.
Abstract: Green's functions play an important role in electroelastic analyses of piezoelectric media. However, most works available on the topic are for the case of uniform temperature. Based on the compact 2-D general solution of orthotropic piezothermoelectric material, which is expressed in harmonic functions, and employing the trial-and-error method, the 2-D Green's function for a steady line heat source in a semi-infinite piezothermoelectric plane is presented by four newly induced harmonic functions. All components of the coupled field are expressed in terms of elementary functions and are convenient to use. Numerical results are given graphically by contours.
TL;DR: In this article, the authors present a study of characteristics for a particle in a damped well, which can be considered as a discretized version of the Melnikov [Phys. Rev. E 48, 3271 (1993)] turnover theory.
Abstract: Since the Kramers problem cannot be, in general, solved in terms of elementary functions, various numerical techniques or approximate methods must be employed. We present a study of characteristics for a particle in a damped well, which can be considered as a discretized version of the Melnikov [Phys. Rev. E 48, 3271 (1993)] turnover theory. The main goal is to justify the direct computational scheme to the basic Wiener-Hopf model. In contrast to the Melnikov approach, which implements factorization through a Cauchy-theorem-based formulation, we employ the Wiener-Levy theorem to reduce the Kramers problem to a Wiener-Hopf sum equation written in terms of Toeplitz matrices. This latter can provide a stringent test for the reliability of analytic approximations for energy distribution functions occurring in the Kramers problems at arbitrary damping. For certain conditions, the simulated characteristics are compared well with those determined using the conventional Fourier-integral formulas, but sometimes may differ slightly depending on the value of a dissipation parameter. Another important feature is that, with our method, we can avoid some complications inherent to the Melnikov method. The calculational technique reported in the present paper may gain particular importance in situations where the energy losses of the particle to the bath are a complex-shaped function of the particle energy and analytic solutions of desired accuracy are not at hand. In order to appreciate more readily the significance and scope of the present numerical approach, we also discuss concrete aspects relating to the field of superionic conductors.
TL;DR: In this paper, the authors proposed an analytical integration of the canonical system by employing its first integrals and invariant expressions, which can be used as reference trajectories for guidance algorithms and to compute initial values of Lagrange multipliers for high-fidelity trajectory optimization software.
Abstract: The variational problem of determining optimal trajectories of motion with constant exhaust velocity and limited mass-flow rate in a central Newtonian field is considered. The first-order necessary conditions of optimality reduce the problem to a Hamiltonian canonical system of equations for intermediate- and maximum-thrust arcs, both of which have no complete analytical solutions to date. The approach used in this work is based on the analytical integration of the canonical system by employing its first integrals and invariant expressions. Several new classes of extremal analytical solutions for planar intermediate-thrust arcs with free and fixed flight times are presented. The solutions describe families of spiral trajectories around the center of attraction. The main result of the paper is that, in their current form with known integrals, the differential equations of the variational problem for intermediate-thrust arcs are integrable in elementary functions and quadratures, and the solution of this problem with such arcs can be reduced to a system of algebraic continuity equations formed for each junction point. These solutions can be used as representative reference trajectories for guidance algorithms and to compute initial values of Lagrange multipliers for high-fidelity trajectory optimization software. As an illustrative example, the transfer maneuver to a given elliptical parking orbit using an intermediate-thrust arc is discussed. Results of simulations for three study cases containing the change of eccentricity and semiparameter of the parking orbit and specific impulses are presented.
TL;DR: In this article, the Green s function of an exponentially graded elastic material in three dimensions was derived by suitably expanding a term in the defining inverse Fourier integral, which can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically.
Abstract: New computational forms are derived for the Green s function of an exponentially graded elastic material in three dimensions. By suitably expanding a term in the defining inverse Fourier integral, the displacement tensor can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically. The integration is over a fixed finite domain, the integrand involves only elementary functions, and only low order Gauss quadrature is required for an accurate answer. Moreover, it is expected that this approach will allow a far simpler procedure for obtaining the first and second order derivatives needed in a boundary integral analysis. The new Green s function expressions have been tested by comparing with results from an earlier algorithm
TL;DR: A computationally efficient technique, based on the method of moments (MoM) formulation, is invoked in the characterization of radiation and scattering properties of an array of coaxial, circular, non-identical loops, applicable to large arrays with respect to diameter or number of loops.
Abstract: A computationally efficient technique, based on the method of moments (MoM) formulation, is invoked in the characterization of radiation and scattering properties of an array of coaxial, circular, non-identical loops. A set of Pocklington-type integral equations for the loop currents is formulated and subsequently discretized by a standard procedure. Thanks to a suitable choice of the basis functions, the resulting matrix corresponding to the pertinent linear system is forced to consist of circulant blocks. This type of system is solvable by an innovative recursive algorithm, featuring several important advantages, such as lower memory and execution time consumption, over standard, purely numerical inversion. The overall procedure is simpler in implementation than already existing methods, based on Fourier analysis. The procedure invokes almost exclusively elementary functions, and is applicable to large arrays with respect to diameter or number of loops. Data for such configurations are presented for the first time in literature.
TL;DR: Fractional exponential that are invariant under fractional derivatives, elementary and special fractional functions are introduced and real integral representations for some H-functions are found that may be very helpful in numerical computations.
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TL;DR: Algorithms for higher order derivative computation of the rectangular QR and eigenvalue decomposition of symmetric matrices with distinct eigenvalues in the forward and reverse mode of algorithmic differentiation using univariate Taylor propagation of matrices (UTPM).
Abstract: We derive algorithms for higher order derivative computation of the rectangular QR and eigenvalue decomposition of symmetric matrices with distinct eigenvalues in the forward and reverse mode of algorithmic differentiation (AD) using univariate Taylor propagation of matrices (UTPM). Linear algebra functions are regarded as elementary functions and not as algorithms. The presented algorithms are implemented in the BSD licensed AD tool ALGOPY. Numerical tests show that the UTPM algorithms derived in this paper produce results close to machine precision accuracy. The theory developed in this paper is applied to compute the gradient of an objective function motivated from optimum experimental design:
TL;DR: In this paper, a stepwise spherically symmetrical potential of interaction was used to obtain the equations for the high-precision description of the second virial coefficient and the equations in "elementary functions" for the compressibility factor of a real gas with nonpolar and polar molecules, describing the thermal properties of substances in a wide range of parameters of state.
Abstract: By means of the stepwise spherically symmetrical potential of interaction, we obtain the equations for the high-precision description of the second virial coefficient and the equations in “elementary functions” for the compressibility factor of a real gas with nonpolar and polar molecules, describing the thermal properties of substances in a wide range of parameters of state with an error close to experimental. It is shown that these equations make it possible to extrapolate with sufficiently high accuracy beyond the limits of the described area. It is established that using these equations, we can calculate the caloric properties of substances with an error, close that of calorimetric experiment.
TL;DR: An explicit representation for the two-dimensional free-surface Green's function in water of infinite depth is derived, based on a finite combination of complex-valued exponential integrals and elementary functions, which can easily and accurately be evaluated in a numerical manner.
Abstract: In this paper we derive an explicit representation for the two-dimensional free-surface Green's function in water of infinite depth, based on a finite combination of complex-valued exponential integrals and elementary functions. This representation can easily and accurately be evaluated in a numerical manner, and its main advantage over other representations lies in its simplicity and in the fact that it can be extended towards the complementary half-plane in a straightforward manner. It seems that this extension has not been studied rigorously until now, and it is required when boundary integral equations are extended in the same way. For the computation of the Green's function, the limiting absorption principle and a partial Fourier transform along the free surface are used. Some of its properties are also discussed, and an expression for its far field is developed, which allows us to state appropriately the involved radiation condition. This Green's function is then used to solve the two-dimensional in...
TL;DR: In this article, the 2D Green's function for a steady line heat source in the interior of two-phase orthotropic electro-magneto-thermo-elastic plane was constructed by ten newly introduced harmonic functions with undetermined constants.
Abstract: We use the 2D general solutions of orthotropic electro-magneto-thermo-elastic material to construct the 2D Green's function for a steady line heat source in the interior of two-phase orthotropic electro-magneto-thermo-elastic plane by ten newly introduced harmonic functions with undetermined constants. The corresponding coupled field can be obtained by substituting these functions into the general solution, and the undetermined constants can be obtained by the continuous conditions and equilibrium conditions. All components of coupled field are expressed in terms of elementary functions and are convenient to use.
TL;DR: In this paper, Hou et al. presented a 3D Green's function for a steady line heat source in a two-phase orthotropic piezothermoelectric plane.
Abstract: Green’s functions play an important role in electroelastic analyses of piezoelectric media. However, most works available on this topic are on case of identical temperature. As a further work of 3D Green’s functions for a two-phase transversely isotropic piezothermoelectric media (Hou, P.F. and Leung, A.Y.T. 2009. ‘‘Three-dimensional Green’s Functions for Two-phase Transversely Isotropic Piezothermoelastic Media,’’ J. Intel. Mater. Syst. Str., 16(5): 1915—1923), the 2D Green’s function for a steady line heat source in a two-phase orthotropic piezothermoelectric plane is presented by eight newly induced harmonic functions in form of elementary functions with undetermined constants. The corresponding coupled field can be obtained by substituting these functions into the general solution, and the undetermined constants can be obtained by the continuous conditions and equilibrium conditions.
TL;DR: In this paper, the authors improved Hubner's sharp upper bound for the Hersch-Pfluger distortion function in quasiconformal theory and provided better estimates for the solutions to the Ramanujan modular equations.
Abstract: New better estimates, which are given in terms of elementary functions, for the function r 7→ (2/�)(1 − r 2 )K(r)K ' (r) + log r appearing in Hubner's sharp upper bound for the Hersch-Pfluger distortion function are obtained. With these estimates, some known bounds for the Hersch-Pfluger distortion function in quasiconformal theory are improved, thus improving the explicit quasiconformal Schwarz lemma and some known estimates for the solutions to the Ramanujan modular equations.
01 Jan 2010
TL;DR: In this paper, a general class of solutions to various partial differential equations known as solitons or stable solitary wave solutions are investigated, which can be written as a polynomial in two elementary functions which satisfy a projective Riccati system.
Abstract: In this paper we investigate a general class of solutions to various partial differential equations known as solitons or stable solitary wave solutions. Many soliton solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective (hence linearizable) Riccati system. From that property, we will use a method for building these solutions by determining only a finite number of coefficients. This method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation. Solutions for the nonlinear equations such as one-dimensional Burgers and KdV-Burgers’ equtions are obtained precisely and so the efficiency of the method can be demonstrated.
TL;DR: A simple Mathematica program for computing the S -state energies and wave functions of two-electron (helium-like) atoms (ions) is presented using the well-known method of projecting the Schrodinger equation onto the finite subspace of basis functions.
TL;DR: In this paper, the Liouville-Neumann expansion at a regular singular point was applied to the confluent and biconfluent Heun's differential equations to approximate the regular solution at the origin.
Abstract: We consider the standard, the confluent and the biconfluent Heun's differential equations. We apply the Liouville–Neumann expansion at a regular singular point introduced in Lopez, [J.L. Lopez, The Liouville–Neumann expansion at a regular singular point, J. Diff. Equ. Appl. 15(2) (2009), pp. 119–132] to approximate the regular solution at the origin of these three differential equations. We design, for each of the three Heun's functions, a sequence of elementary functions (rational functions and polylogarithms) that converges uniformly over compacts to the given Heun's function. This sequence (Liouville–Neumann expansion) is defined by means of an integral recurrence. Some numerical experiments show that the convergence of the Liouville–Neumann expansion is extraordinarily fast.