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1

Homotopy perturbation Shehu transform method for solving fractional models arising in applied sciences

Maitama Shehu, Zhao Weidong

Journal of Applied Mathematics and Computational Mechanics

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2021

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Vol. 20, nr 1

71--82

EN

Using the recently proposed homotopy perturbation Shehu transform method (HPSTM), we successfully construct reliable solutions of some important fractional models arising in applied physical sciences. The nonlinear terms are decomposed using He’s polynomials, and the fractional derivative is calculated in the Caputo sense. Using the analytical method, we obtained the exact solution of the fractional diffusion equation, fractional wave equation and the nonlinear fractional gas dynamic equation.

2

Application of Residual Power Series Method for the Solution of Time-fractional Schrödinger Equations in One-dimensional Space

Arqub Omar Abu

Fundamenta Informaticae

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2019

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Vol. 166, nr 2

87--110

EN

The object of this article is to present the computational solution of the time-fractional Schrödinger equation subject to given constraint condition based on the generalized Taylor series formula in the Caputo sense. The algorithm methodology is based on construct a multiple fractional power series solution in the form of a rabidly convergent series with minimum size of calculations using symbolic computation software. The proposed technique is fully compatible with the complexity of this problem and obtained results are highly encouraging. Efficacious computational experiments are provided to guarantee the procedure and to illustrate the theoretical statements of the present algorithm in order to show its potentiality, generality, and superiority for solving such fractional equation. Graphical results and numerical comparisons are presented and discussed quantitatively to illustrate the solution.

3

Symbolic approach to the general cubic decomposition of polynomial sequences. Results for several orthogonal and symmetric cases

Mesquita T. A., Rocha Z. Da

Opuscula Mathematica

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2012

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Vol. 32, no. 4

675-687

EN

We deal with a symbolic approach to the cubic decomposition (CD) of polynomial sequences - presented in a previous article referenced herein - which allows us to compute explicitly the first elements of the nine component sequences of a CD. Properties are investigated and several experimental results are discussed, related to the CD of some widely known orthogonal sequences. Results concerning the symmetric character of the component sequences are established.

4

Computational Solutions for the Korteweg–deVries Equation in Warm Plasma

El-Shewy E. K., Abdelwahed H. G., Abd-El-Hamid H. M.

Computational Methods in Science and Technology

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2010

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Vol. 16, No. 1

13-18

EN

The reductive perturbation method has been employed to derive the Korteweg–deVries (KdV) equation for small but finite amplitude electrostatic ion-acoustic waves. An algebraic method with computerized symbolic computation, which greatly exceeds the applicability of the existing tanh, extended tanh methods in obtaining a series of exact solutions of the KdV equation. Numerical studies have been made using plasma parameters to reveal different solutions, i.e. bell-shaped solitary pulses, rational pulses and solutions with singularity at a finite points called “blowup” solutions in addition to the propagation of explosive pulses. The result of the present investigation may be applicable to some plasma environments, such as ionosphere.

5

Zagadnienia wrażliwości własności efektywnych materiałów kompozytowych

Kamiński M.

Kompozyty

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2003

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R. 3, nr 7

232-237

PL

Teoretyczne i numeryczne aspekty wrażliwości własności efektywnych materiałów kompozytowych o składnikach liniowo-spreżystych i poprzecznie izotropowych są tematem niniejszej pracy. Zostały one przedyskutowane i zilustrowane przykładami inżynierskimi dotyczącymi kompozytów periodycznych i dwuskładnikowych o osiowo rozłożonych niejednorodnościach oraz struktur włóknistych, jednakże model ogólny został sformułowany dla kompozytów o dowolnej liczbie składników. Obliczenia numeryczne zostały wykonane za pomocą systemu komputerowego MAPLE do rachunku symbolicznego, a także wykorzystując specjalnie zmodyfikowany program Metody Elementów Skończonych MCCEFF. W celu zweryfikowania wrażliwości różnych kompozytów na ogólny stan odkształcenia/naprężenia wprowadzono funkcjonał wrażliwości oparty na energii odkształcenia w jednostkowym stanie równomiernego osiowego i postaciowego odkształcenia komórki periodyczności. Dalsze analizy numeryczne w tym zakresie pozwolą na analizę wrażliwości innych materiałów kompozytowych, optymalizacją różnych kompozytów, rozwiązywanie zagadnień identyfikacji w kompozytach, jak również analizę wrażliwości na kształt struktur ujednorodnionych oraz modelowanie wrażliwości materiałów kompozytowych o własnościach losowych.

EN

The main aim is to present theoretical aspects and the Finite Element Method implementation of sensitivity analysis in homogenization of composite materials with linear elastic components using the effective modules theory. Sensitivity analysis of effective material properties is presented in a general form for n-component periodic composite and is illustrated by the examples of periodic 1D as well as 2D heterogeneous structures; sensitivity coefficients are determined for the effective elasticity tensor components. The structural response functional for the fiber-reinforced elastic composite is proposed in a form of total strain energy resulting from some uniform strain state imposed on the composite Representative Volume Element (RVE). Numerical results confirm the usefulness of the homogenization method in computational analysis of composite materials sensitivity and may be applied in composite optimization, identification, shape sensitivity studies as well as in stochastic sensitivity analysis of random composites.

6

Przekształcenia symboliczne w modelowaniu dynamiki mobilnych robotów kołowych

Żylski W., Buratowski T.

Zeszyty Naukowe Katedry Mechaniki Stosowanej / Politechnika Śląska

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2001

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z. 15

359-364

PL

W pracy dokonano analizy opisu ruchu dwukołowego mobilnego robota Pioneer-2DX. Założono, że mobilny robot przemieszcza się w płaszczyźnie xy. Do opisu dynamiki tego nieholonomicznego układu można stosować równania Lagrange'a II rodzaju z mnożnikami oraz równania Maggiego. W artykule przedstawiono dynamiczne równania ruchu stosując równania Maggiego.

EN

In this paper the problem concerning dynamics of wheeled mobile robot Pioneer-2DX and is discussed. The motion of the analysed model can be described in xy plane. Taking account of equations of non-holonomic constraints, motion phenomenon is described using Lagrange equations of the II-nd type with multipliers and Maggi equations. The motion of the analysed model is described by system of differential equations, based on Maggi equations.

7

Symbolic computations in structural dynamics using stochastic second order perturbation method

Kamiński M.

Mechanics and Mechanical Engineering

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2001

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Vol. 5, nr 1

69-80

EN

The main idea is to show the application of symbolic computations in analysis of engineering systems with random parameters. The general computational methodology is based on the stochastic second order perturbation method and its implementation in the mathematical package MAPLE. The entire approach is displayed on the example of a simple single degree of freedom dynamical system with random spring stiffness. The results of symbolic computations are derived numerically in the form of probabilistic moments of the structural response, computed for the whole analysis time domain. This methodology can be applied to all the engineering problems, where the response can be derived symbolically in the deterministic case, while input parameters of the system are random variables, fields or processes characterized by the probability density function (PDF) of any type.