13 results
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2. A new modification into Quasi-Newton equation for solving unconstrained optimization problems.
- Author
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Hassan, Basim A., Taha, Mohammed W., Kadoo, Farah H., and Mohammed, Sulaiman Ibrahim
- Subjects
QUASI-Newton methods ,NUMERICAL analysis ,EQUATIONS - Abstract
One of the most common problems with quasi-Newton methods is how to select the suitable quasi-Newton equation while operating the algorithm. In this paper, a new modification Quasi-Newton equation is proposed based on recent quasi-Newton equation of the Biglari et.al. method. Theoretical analyses and numerical computations using standard functions, as well as comparison with Broyden-Fletcher-Goldfarb-Shanno (BFGS) show that the proposed new type is globally convergent and computationally efficient. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. On the proper derivation of the Floquet-based quantum classical Liouville equation and surface hopping describing a molecule or material subject to an external field.
- Author
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Chen, Hsing-Ta, Zhou, Zeyu, and Subotnik, Joseph E.
- Subjects
NUMERICAL analysis ,DEGREES of freedom ,EQUATIONS ,HOPS ,MOLECULES - Abstract
We investigate different approaches to derive the proper Floquet-based quantum–classical Liouville equation (F-QCLE) for laser-driven electron-nuclear dynamics. The first approach projects the operator form of the standard QCLE onto the diabatic Floquet basis and then transforms to the adiabatic representation. The second approach directly projects the QCLE onto the Floquet adiabatic basis. Both approaches yield a form that is similar to the usual QCLE with two modifications: (1) The electronic degrees of freedom are expanded to infinite dimension and (2) the nuclear motion follows Floquet quasi-energy surfaces. However, the second approach includes an additional cross derivative force due to the dual dependence on time and nuclear motion of the Floquet adiabatic states. Our analysis and numerical tests indicate that this cross derivative force is a fictitious artifact, suggesting that one cannot safely exchange the order of Floquet state projection with adiabatic transformation. Our results are in accord with similar findings by Izmaylov et al., [J. Chem. Phys. 140, 084104 (2014)] who found that transforming to the adiabatic representation must always be the last operation applied, although now we have extended this result to a time-dependent Hamiltonian. This paper and the proper derivation of the F-QCLE should lay the basis for further improvements of Floquet surface hopping. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
4. Higher order difference numerical analyses of a 2D Poisson equation by the interpolation finite difference method and calculation error evaluation.
- Author
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Fukuchi, Tsugio
- Subjects
FINITE difference method ,NUMERICAL analysis ,INTERPOLATION ,EQUATIONS ,POISEUILLE flow - Abstract
In a previous paper, a calculation system for a high-accuracy, high-speed calculation of a one-dimensional (1D) Poisson equation based on the interpolation finite difference method was shown. Spatial high-order finite difference (FD) schemes, including a usual second-order accurate centered space FD scheme, are instantaneously derived on the equally spaced/unequally spaced grid points based on the definition of the Lagrange polynomial function. The upper limit of the higher order FD scheme is not theoretically limited but is studied up to the tenth order, following the previous paper. In the numerical analyses of the 1D Poisson equation published in the previous paper, the FD scheme setting method, SAPI (m), m = 2, 4, ..., 10, was defined. Due to specifying the value of m, the setting of FD schemes is uniquely defined. This concept is extended to the numerical analysis of two-dimensional Poisson equations. In this paper, we focus on Poiseuille flows passing through arbitrary cross sections as numerical calculation examples. Over regular and irregular domains, three types of FD methods—(i) forward time explicit method, (ii) time marching successive displacement method, and (iii) alternative direction implicit method—are formulated, and their characteristics of convergence and numerical calculation errors are investigated. The numerical calculation system of the 2D Poisson equation formulated in this paper enables high-accuracy and high-speed calculation by the high-order difference in an arbitrary domain. Especially in the alternative direction implicit method using the band diagonal matrix algorithm, convergence is remarkably accelerated, and high-speed calculation becomes possible. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
5. Special Second Order Stiff Delay Differential Equations Directly Solved by Using Variable Stepsize Variable Order Technique.
- Author
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Mohd Isa, Nora Baizura and Ishak, Fuziyah
- Subjects
DELAY differential equations ,MATHEMATICAL variables ,NUMERICAL solutions to differential equations ,NUMERICAL analysis ,EQUATIONS - Abstract
Less attention was made to solve special second order stiff delay differential equations (DDEs) directly. In this paper, the development of direct predictor-corrector variable stepsize variable order (DBVSVO) method is described to solve special second order stiff DDEs directly without reducing to first order equations. The predictor and corrector formulae is derived based on backward differentiation formulae (BDF) and represented in divided difference form. In order to achieve greater efficiency, the developed method is implemented using variable stepsize variable order (VSVO) technique. Numerical results are presented to show that the developed method is suitable for solving special second order stiff DDEs. For comparison purposes the same set of test examples are reduced to a system of the first order equations and solved using predictor-corrector variable stepsize variable order method based on backward differentiation formulae (BVSVO) method proposed by Mohd Isa et al. [1]. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
6. On Genocchi Operational Matrix of Fractional Integration for Solving Fractional Differential Equations.
- Author
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Abdulnasir Isah and Chang Phang
- Subjects
FRACTIONAL integrals ,MATHEMATICS ,POLYNOMIALS ,MATHEMATICAL analysis ,NUMERICAL analysis ,EQUATIONS ,ALGEBRA - Abstract
In this paper we present a new numerical method for solving fractional differential equations (FDEs) based on Genocchi polynomials operational matrix through collocation method. The operational matrix of fractional integration in Riemann-Liouville sense is derived. The upper bound for the error of the operational matrix of fractional integration is also shown. The properties of Genocchi polynomials are utilized to reduce the given problems to a system of algebraic equations. Illustrative examples are finally given to show the simplicity, accuracy and applicability of the method. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
7. High-order accurate and high-speed calculation system of 1D Laplace and Poisson equations using the interpolation finite difference method.
- Author
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Fukuchi, Tsugio
- Subjects
FINITE difference method ,INTERPOLATION ,NUMERICAL calculations ,NUMERICAL analysis ,EQUATIONS ,LAPLACE transformation - Abstract
Among the methods of the numerical analysis of the physical phenomena of the continuum, the finite difference method (FDM) is the first examined method and has been established as a full numerical calculation system over the regular domain. However, there is a general perception that generality in numerical calculations cannot be expected over complex irregular domains. As using the FDM, the development of computational methods that are applicable over any irregular domain is considered to be a very important contemporary problem. In the FDM, there is a marked characteristic that the theory developed by the (spatial) one-dimensional (1D) problem is naturally applied to the 2D and 3D problems. The calculation method is called the interpolation FDM (IFDM). In this paper, attention is paid to 1D Laplace and Poisson equations, and the whole image of the IFDM using the algebraic polynomial interpolation method (APIM), the IFDM-APIM, is described. Based on the Lagrange interpolation function, the spatial difference schemes from 2nd order to 10th order including odd order are calculated and defined instantaneously over equally/unequally spaced grid points, then, high-order accurate and high-speed computations become possible. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
8. Physics-based analytical model for ferromagnetic single electron transistor.
- Author
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Jamshidnezhad, K. and Sharifi, M. J.
- Subjects
FERROMAGNETIC materials ,SINGLE electron transistors ,MONTE Carlo method ,EQUATIONS ,NUMERICAL analysis - Abstract
A physically based compact analytical model is proposed for a ferromagnetic single electron transistor (FSET). This model is based on the orthodox theory and solves the master equation, spin conservation equation, and charge neutrality equation simultaneously. The model can be applied to both symmetric and asymmetric devices and does not introduce any limitation on the applied bias voltages. This feature makes the model suitable for both analog and digital applications. To verify the accuracy of the model, its results regarding a typical FSET in both low and high voltage regimes are compared with the existing numerical results. Moreover, the model's results of a parallel configuration FSET, where no spin accumulation exists in the island, are compared with the results obtained from a Monte Carlo simulation using SIMON. These two comparisons show that our model is valid and accurate. As another comparison, the model is compared analytically with an existing model for a double barrier ferromagnetic junction (having no gate). This also verifies the accuracy of the model. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
9. Comparative study of fractional Newell–Whitehead–Segel equation using optimal auxiliary function method and a novel iterative approach.
- Author
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Xin, Xiao, khan, Ibrar, Ganie, Abdul Hamid, Akgül, Ali, Bonyah, Ebenezer, Fathima, Dowlath, and Ali Yousif, Badria Almaz
- Subjects
CAPUTO fractional derivatives ,PARTIAL differential equations ,NUMERICAL analysis ,EQUATIONS ,COMPARATIVE studies - Abstract
This research explores the solution of the time-fractional Newell–Whitehead–Segel equation using two separate methods: the optimal auxiliary function method and a new iterative method. The Newell–Whitehead–Segel equation holds significance in modeling nonlinear systems, particularly in delineating stripe patterns within two-dimensional systems. Employing the Caputo fractional derivative operator, we address two case study problems pertaining to this equation through our proposed methods. Comparative analysis between the numerical results obtained from our techniques and an exact solution reveals a strong alignment. Graphs and tables illustrate this alignment, showcasing the effectiveness of our methods. Notably, as the fractional orders vary, the results achieved at different fractional orders are compared, highlighting their convergence toward the exact solution as the fractional order approaches an integer. Demonstrating both interest and simplicity, our proposed methods exhibit high accuracy in resolving diverse nonlinear fractional order partial differential equations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
10. A New Three-Level Rotated Implicit Method for Solving the Two-Dimensional Time Fractional Diffusion-Wave Equation.
- Author
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Balasim, Alla Tareq and Mohd. Ali, Norhashidah Hj.
- Subjects
- *
DIFFUSION , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis , *ITERATIVE decoding , *EQUATIONS - Abstract
Fractional diffusion-wave equation represents many physical phenomena in modeling diffusive processes, waves in fluid flow, and oil strata among others. The numerical solution of this equation is an important task and has been investigated extensively over the last several years. The main purpose of this paper is to formulate a new three time level method in solving the two dimensional time-fractional diffusion-wave equation based on a rotated finite difference approximation formula where the time fractional derivative is described by Caputo's derivative of order 1 <α < 2 . The developed scheme is derived from the standard implicit formula rotated 45° clockwise with respect to the standard mesh. Numerical example and comparison with the standard classical iterative method has been conducted in this study to test the effectiveness of the proposed method. We show that the proposed iterative method is superior to the standard iterative method in terms of iteration numbers and execution timings without having to jeopardize the accuracy of the solutions. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
11. Bumps and oscillons in networks of spiking neurons.
- Author
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Schmidt, Helmut and Avitabile, Daniele
- Subjects
ACTION potentials ,GALERKIN methods ,NUMERICAL analysis ,OSCILLATIONS ,EQUATIONS - Abstract
We study localized patterns in an exact mean-field description of a spatially extended network of quadratic integrate-and-fire neurons. We investigate conditions for the existence and stability of localized solutions, so-called bumps, and give an analytic estimate for the parameter range, where these solutions exist in parameter space, when one or more microscopic network parameters are varied. We develop Galerkin methods for the model equations, which enable numerical bifurcation analysis of stationary and time-periodic spatially extended solutions. We study the emergence of patterns composed of multiple bumps, which are arranged in a snake-and-ladder bifurcation structure if a homogeneous or heterogeneous synaptic kernel is suitably chosen. Furthermore, we examine time-periodic, spatially localized solutions (oscillons) in the presence of external forcing, and in autonomous, recurrently coupled excitatory and inhibitory networks. In both cases, we observe period-doubling cascades leading to chaotic oscillations. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
12. Geometry and Symmetry in Non-equilibrium Thermodynamic Systems.
- Author
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Sonnino, Giorgio
- Subjects
THERMODYNAMIC equilibrium ,GEOMETRY ,EQUATIONS ,NUMERICAL analysis ,ENTROPY - Abstract
The ultimate aim of this series of works is to establish the closure equations, valid for thermodynamic systems out from the Onsager region, and to describe the geometry and symmetry in thermodynamic systems far from equilibrium. Geometry of a non-equilibrium thermodynamic system is constructed by taking into account the second law of thermodynamics and by imposing the validity of the Glansdorff-Prigogine Universal Criterion of Evolution. These two constraints allow introducing the metrics and the affine connection of the Space of the Thermodynamic Forces, respectively. The Lie group associated to the nonlinear Thermodynamic Coordinate Transformations (TCT) leaving invariant both the entropy production σ and the Glansdorff-Prigogine dissipative quantity P, is also described. The invariance under TCT leads to the formulation of the Thermodynamic Covariance Principle (TCP): The nonlinear closure equations, i.e. the flux-force relations, must be covariant under TCT. In other terms, the fundamental laws of thermodynamics should be manifestly covariant under transformations between the admissible thermodynamic forces (i.e. under TCT). The symmetry properties of a physical system are intimately related to the conservation laws characterizing the thermodynamic system. Noether's theorem gives a precise description of this relation. The macroscopic theory for closure relations, based on this geometrical description and subject to the TCP, is referred to as the Thermodynamic Field Theory (TFT). This theory ensures the validity of the fundamental theorems for systems far from equilibrium. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
13. New Types of Multisoliton Solutions of Some Integrable Equations via Direct Methods.
- Author
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Burde, Georgy I.
- Subjects
NUMERICAL analysis ,EQUATIONS ,SOLITONS ,GEOMETRIC connections ,HYPERBOLIC functions - Abstract
Exact explicit solutions, which describe new multisoliton dynamics, have been identified for some KdV type equations using direct methods devised for this purpose. It is found that the equations, having multi-soliton solutions in terms of the KdV-type solitons, possess also an alternative set of multi-soliton solutions which include localized static structures that behave like (static) solitons when they collide with moving solitons. The alternative sets of solutions include the steadystate solution describing the static soliton itself and unsteady solutions describing mutual interactions in a system consisting of a static soliton and several moving solitons. As distinct from common multisoliton solutions those solutions represent combinations of algebraic and hyperbolic functions and cannot be obtained using the traditional methods of soliton theory. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
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