Theory of Stability and Convergence

2006 ◽  
pp. 327-399
Keyword(s):  
Stability And Convergence

scholarly journals An attractive numerical algorithm for solving nonlinear Caputo–Fabrizio fractional Abel differential equation in a Hilbert space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed Al-Smadi ◽  
Nadir Djeddi ◽  
Shaher Momani ◽  
Shrideh Al-Omari ◽  
Serkan Araci
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Reproducing Kernel ◽  
Numerical Approach ◽  
Accurate Solution ◽  
Stability And Convergence ◽  
Type Model ◽  
Schmidt Orthogonalization ◽  
And Performance ◽  
Abel Differential Equation

AbstractOur aim in this paper is presenting an attractive numerical approach giving an accurate solution to the nonlinear fractional Abel differential equation based on a reproducing kernel algorithm with model endowed with a Caputo–Fabrizio fractional derivative. By means of such an approach, we utilize the Gram–Schmidt orthogonalization process to create an orthonormal set of bases that leads to an appropriate solution in the Hilbert space $\mathcal{H}^{2}[a,b]$ H 2 [ a , b ] . We investigate and discuss stability and convergence of the proposed method. The n-term series solution converges uniformly to the analytic solution. We present several numerical examples of potential interests to illustrate the reliability, efficacy, and performance of the method under the influence of the Caputo–Fabrizio derivative. The gained results have shown superiority of the reproducing kernel algorithm and its infinite accuracy with a least time and efforts in solving the fractional Abel-type model. Therefore, in this direction, the proposed algorithm is an alternative and systematic tool for analyzing the behavior of many nonlinear temporal fractional differential equations emerging in the fields of engineering, physics, and sciences.

scholarly journals Entropy Generation Analysis and Radiated Heat Transfer in MHD (Al2O3-Cu/Water) Hybrid Nanofluid Flow

Micromachines ◽  
2021 ◽  
Vol 12 (8) ◽  
pp. 887
Author(s):  
Nabeela Parveen ◽  
Muhammad Awais ◽  
Saeed Ehsan Awan ◽  
Wasim Ullah Khan ◽  
Yigang He ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Entropy Generation ◽  
Magnetic Induction ◽  
Axisymmetric Flow ◽  
Computational Method ◽  
Stability And Convergence ◽  
Hybrid Nanofluid ◽  
Physical Factors ◽  
Induction Effect ◽  
Entropy Generation Number

This research concerns the heat transfer and entropy generation analysis in the MHD axisymmetric flow of Al2O3-Cu/H2O hybrid nanofluid. The magnetic induction effect is considered for large magnetic Reynolds number. The influences of thermal radiations, viscous dissipation and convective temperature conditions over flow are studied. The problem is modeled using boundary layer theory, Maxwell’s equations and Fourier’s conduction law along with defined physical factors. Similarity transformations are utilized for model simplification which is analytically solved with the homotopy analysis method. The h-curves upto 20th order for solutions establishes the stability and convergence of the adopted computational method. Rheological impacts of involved parameters on flow variables and entropy generation number are demonstrated via graphs and tables. The study reveals that entropy in system of hybrid nanofluid affected by magnetic induction declines for [...]

scholarly journals A Hybridization of Dragonfly Algorithm Optimization and Angle Modulation Mechanism for 0-1 Knapsack Problems

Entropy ◽  
2021 ◽  
Vol 23 (5) ◽  
pp. 598
Author(s):  
Lin Wang ◽  
Ronghua Shi ◽  
Jian Dong
Keyword(s):  
Vertical Displacement ◽  
Continuous Functions ◽  
Convergence Speed ◽  
Knapsack Problems ◽  
Stability And Convergence ◽  
Algorithm Optimization ◽  
Solution Quality ◽  
Dragonfly Algorithm ◽  
Algorithm Stability ◽  
Angle Modulation

The dragonfly algorithm (DA) is a new intelligent algorithm based on the theory of dragonfly foraging and evading predators. DA exhibits excellent performance in solving multimodal continuous functions and engineering problems. To make this algorithm work in the binary space, this paper introduces an angle modulation mechanism on DA (called AMDA) to generate bit strings, that is, to give alternative solutions to binary problems, and uses DA to optimize the coefficients of the trigonometric function. Further, to improve the algorithm stability and convergence speed, an improved AMDA, called IAMDA, is proposed by adding one more coefficient to adjust the vertical displacement of the cosine part of the original generating function. To test the performance of IAMDA and AMDA, 12 zero-one knapsack problems are considered along with 13 classic benchmark functions. Experimental results prove that IAMDA has a superior convergence speed and solution quality as compared to other algorithms.

scholarly journals Randomized Runge–Kutta method — Stability and convergence under inexact information

2021 ◽  
pp. 101554
Author(s):  
Tomasz Bochacik ◽  
Maciej Goćwin ◽  
Paweł M. Morkisz ◽  
Paweł Przybyłowicz
Keyword(s):  
Stability And Convergence ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Inexact Information ◽  
Runge Kutta

scholarly journals A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Muhammad Asim Khan ◽  
Norhashidah Hj. Mohd Ali ◽  
Nur Nadiah Abd Hamid
Keyword(s):  
Finite Difference ◽  
Stokes Problem ◽  
Stability And Convergence ◽  
Second Grade ◽  
Group Method ◽  
Two Dimensional ◽  
Second Grade Fluid ◽  
The Matrix ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid

Abstract In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

scholarly journals Numerical simulation of periodic MHD casson nanofluid flow through porous stretching sheet

2021 ◽  
Vol 3 (2) ◽  
Author(s):  
Abdullah Al-Mamun ◽  
S. M. Arifuzzaman ◽  
Sk. Reza-E-Rabbi ◽  
Umme Sara Alam ◽  
Saiful Islam ◽  
...  
Keyword(s):  
Fluid Flow ◽  
Lewis Number ◽  
Stability And Convergence ◽  
Radiation Absorption ◽  
Physical Parameters ◽  
Theoretical Idea ◽  
Prostate Cancer Therapy ◽  
Flow Through ◽  
Explicit Finite Difference ◽  
The Stability

AbstractThe perspective of this paper is to characterize a Casson type of Non-Newtonian fluid flow through heat as well as mass conduction towards a stretching surface with thermophoresis and radiation absorption impacts in association with periodic hydromagnetic effect. Here heat absorption is also integrated with the heat absorbing parameter. A time dependent fundamental set of equations, i.e. momentum, energy and concentration have been established to discuss the fluid flow system. Explicit finite difference technique is occupied here by executing a procedure in Compaq Visual Fortran 6.6a to elucidate the mathematical model of liquid flow. The stability and convergence inspection has been accomplished. It has observed that the present work converged at, Pr ≥ 0.447 indicates the value of Prandtl number and Le ≥ 0.163 indicates the value of Lewis number. Impact of useful physical parameters has been illustrated graphically on various flow fields. It has inspected that the periodic magnetic field has helped to increase the interaction of the nanoparticles in the velocity field significantly. The field has been depicted in a vibrating form which is also done newly in this work. Subsequently, the Lorentz force has also represented a great impact in the updated visualization (streamlines and isotherms) of the flow field. The respective fields appeared with more wave for the larger values of magnetic parameter. These results help to visualize a theoretical idea of the effect of modern electromagnetic induction use in industry instead of traditional energy sources. Moreover, it has a great application in lung and prostate cancer therapy.

Control of Fractional-Order Systems Using Chatter-Free Sliding Mode Approach

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Mohammad Pourmahmood Aghababa
Keyword(s):  
Fractional Derivative ◽  
Sliding Mode ◽  
Variable Structure ◽  
Stability And Convergence ◽  
Control Input ◽  
Fractional Systems ◽  
Sliding Motion ◽  
Efficient Performance ◽  
Sliding Manifold ◽  
Chattering Free

The problem of stabilization of nonlinear fractional systems in spite of system uncertainties is investigated in this paper. First, a proper fractional derivative type sliding manifold with desired stability and convergence properties is designed. Then, the fractional stability theory is adopted to derive a robust sliding control law to force the system trajectories to attain the proposed sliding manifold and remain on it evermore. The existence of the sliding motion is mathematically proven. Furthermore, the sign function in the control input, which is responsible to the being of harmful chattering, is transferred into the fractional derivative of the control input. Therefore, the resulted control input becomes smooth and free of the chattering. Some numerical simulations are presented to illustrate the efficient performance of the proposed chattering-free fractional variable structure controller.

An Efficient Galerkin BEM to Compute High Acoustic Eigenfrequencies

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Mario Durán ◽  
Jean-Claude Nédélec ◽  
Sebastián Ossandón
Keyword(s):  
Numerical Method ◽  
Integral Equations ◽  
High Frequency ◽  
High Accuracy ◽  
Integral Representations ◽  
Stability And Convergence ◽  
Symmetric Matrices ◽  
Frequency Interval ◽  
Numerical Treatment ◽  
The Stability

An efficient numerical method, using integral equations, is developed to calculate precisely the acoustic eigenfrequencies and their associated eigenvectors, located in a given high frequency interval. It is currently known that the real symmetric matrices are well adapted to numerical treatment. However, we show that this is not the case when using integral representations to determine with high accuracy the spectrum of elliptic, and other related operators. Functions are evaluated only in the boundary of the domain, so very fine discretizations may be chosen to obtain high eigenfrequencies. We discuss the stability and convergence of the proposed method. Finally we show some examples.

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