1,871 research outputs found
A semi–analytical study of diffusion type multi–term time fractional partial differential equation
This work suggested algorithm for the solution of multi–term time fractional partial differential equation by the application of homotopy analysis fractional Sumudu transform method based on iterative process. The method is cumulation of Sumudu transform and homotopy analysis method. Also, the multi-term time fractional partial differential equation represented in the form of system of fractional partial differential equations as per certain conditions of fractional derivatives. The Caputo fractional order derivatives are taken for the multi–term time fractional partial differential equations. Numerical examples are discussed for the support of theory and the approximate solution compared with exact solutions at the integer value of derivatives.Emerging Sources Citation Index (ESCI)MathScinetScopu
Numerical solution of fractional partial differential equations by spectral methods
Fractional partial differential equations (FPDEs) have become essential tool for the modeling of physical models by using spectral methods. In the last few decades, spectral methods have been developed for the solution of time and space dimensional FPDEs. There are different types of spectral methods such as collocation methods, Tau methods and Galerkin methods. This research work focuses on the collocation and Tau methods to propose an efficient operational matrix methods via Genocchi polynomials and Legendre polynomials for the solution of two and three dimensional FPDEs. Moreover, in this study, Genocchi wavelet-like basis method and Genocchi polynomials based Ritz- Galerkin method have been derived to deal with FPDEs and variable- order FPDEs. The reason behind using the Genocchi polynomials is that, it helps to generate functional expansions with less degree and small coefficients values to derive the operational matrix of derivative with less computational complexity as compared to Chebyshev and Legendre Polynomials. The results have been compared with the existing methods such as Chebyshev wavelets method, Legendre wavelets method, Adomian decomposition method, Variational iteration method, Finite difference method and Finite element method. The numerical results have revealed that the proposed methods have provided the better results as compared to existing methods due to minimum computational complexity of derived operational matrices via Genocchi polynomials. Additionally, the significance of the proposed methods has been verified by finding the error bound, which shows that the proposed methods have provided better approximation values for under consideration FPDEs
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
On a multiwavelet spectral element method for integral equation of a generalized Cauchy problem
In this paper we deal with construction and analysis of a multiwavelet spectral element scheme for a generalized Cauchy type problem with Caputo fractional derivative. Numerical schemes for this type of problems, often suffer from the draw-back of spurious oscillations. A common remedy is to render the problem to an equivalent integral equation. For the generalized Cauchy type problem, a corresponding integral equation is of nonlinear Volterra type. In this paper we investigate wellposedness and convergence of a stabilizing multiwavelet scheme for a, one-dimensional case (in [a,\ua0b] or [0,\ua01]), of this problem. Based on multiwavelets, we construct an approximation procedure for the fractional integral operator that yields a linear system of equations with sparse coefficient matrix. In this setting, choosing an appropriate threshold, the number of non-zero coefficients in the system is substantially reduced. A severe obstacle in the convergence analysis is the lack of continuous derivatives in the vicinity of the inflow/ starting boundary point. We overcome this issue through separating a J (mesh)-dependent, small, neighborhood of a (or origin) from the interval, where we only take L2-norm. The estimate in this part relies on Chebyshev polynomials, viz. As reported by Richardson(Chebyshev interpolation for functions with endpoint singularities via exponential and double-exponential transforms, Oxford University, UK, 2012) and decreases, almost, exponentially by raising J. At the remaining part of the domain the solution is sufficiently regular to derive the desired optimal error bound. We construct such a modified scheme and analyze its wellposedness, efficiency and accuracy. The robustness of the proposed scheme is confirmed implementing numerical examples
A Multiscale Guide to Brownian Motion
We revise the Levy's construction of Brownian motion as a simple though still
rigorous approach to operate with various Gaussian processes. A Brownian path
is explicitly constructed as a linear combination of wavelet-based "geometrical
features" at multiple length scales with random weights. Such a wavelet
representation gives a closed formula mapping of the unit interval onto the
functional space of Brownian paths. This formula elucidates many classical
results about Brownian motion (e.g., non-differentiability of its path),
providing intuitive feeling for non-mathematicians. The illustrative character
of the wavelet representation, along with the simple structure of the
underlying probability space, is different from the usual presentation of most
classical textbooks. Similar concepts are discussed for fractional Brownian
motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional
Gaussian fields. Wavelet representations and dyadic decompositions form the
basis of many highly efficient numerical methods to simulate Gaussian processes
and fields, including Brownian motion and other diffusive processes in
confining domains
A novel design of fractional Mayer wavelet neural networks with application to the nonlinear singular fractional Lane-Emden systems
In this study, a novel stochastic computational frameworks based on fractional Meyer wavelet artificial neural network (FMW-ANN) is designed for nonlinear-singular fractional Lane-Emden (NS-FLE) differential equation. The modeling strength of FMW-ANN is used to transformed the differential NS-FLE system to difference equations and approximate theory is implemented in mean squared error sense to develop a merit function for NS-FLE differential equations. Meta-heuristic strength of hybrid computing by exploiting global search efficacy of genetic algorithms (GA) supported with local refinements with efficient active-set (AS) algorithm is used for optimization of design variables FMW-ANN., i.e., FMW-ANN-GASA. The proposed FMW-ANN-GASA methodology is implemented on NS-FLM for six different scenarios in order to exam the accuracy, convergence, stability and robustness. The proposed numerical results of FMW-ANN-GASA are compared with exact solutions to verify the correctness, viability and efficacy. The statistical observations further validate the worth of FMW-ANN-GASA for the solution of singular nonlinear fractional order systems.This paper is partially supported by Ministerio de Ciencia, Innovación y Universidades grant number PGC2018-097198-BI00
and Fundación Séneca de la Región de Murcia grant number 20783/PI/18
Three real-space discretization techniques in electronic structure calculations
A characteristic feature of the state-of-the-art of real-space methods in
electronic structure calculations is the diversity of the techniques used in
the discretization of the relevant partial differential equations. In this
context, the main approaches include finite-difference methods, various types
of finite-elements and wavelets. This paper reports on the results of several
code development projects that approach problems related to the electronic
structure using these three different discretization methods. We review the
ideas behind these methods, give examples of their applications, and discuss
their similarities and differences.Comment: 39 pages, 10 figures, accepted to a special issue of "physica status
solidi (b) - basic solid state physics" devoted to the CECAM workshop "State
of the art developments and perspectives of real-space electronic structure
techniques in condensed matter and molecular physics". v2: Minor stylistic
and typographical changes, partly inspired by referee comment
Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets
This article describes an approximation technique based on fractional order
Bernstein wavelets for the numerical simulations of fractional oscillation
equations under variable order, and the fractional order Bernstein wavelets are
derived by means of fractional Bernstein polynomials. The oscillation equation
describes electrical circuits and exhibits a wide range of nonlinear dynamical
behaviors. The proposed variable order model is of current interest in a lot of
application areas in engineering and applied sciences. The purpose of this
study is to analyze the behavior of the fractional force-free and forced
oscillation equations under the variable-order fractional operator. The basic
idea behind using the approximation technique is that it converts the proposed
model into non-linear algebraic equations with the help of collocation nodes
for easy computation. Different cases of the proposed model are examined under
the selected variable order parameters for the first time in order to show the
precision and performance of the mentioned scheme. The dynamic behavior and
results are presented via tables and graphs to ensure the validity of the
mentioned scheme. Further, the behavior of the obtained solutions for the
variable order is also depicted. From the calculated results, it is observed
that the mentioned scheme is extremely simple and efficient for examining the
behavior of nonlinear random (constant or variable) order fractional models
occurring in engineering and science.Comment: This is a preprint of a paper whose final and definite form is
published Open Access in 'Mathematics' at
[http://dx.doi.org/10.3390/math11112503
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