A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

Raimondas Čiegis; Remigijus Čiegis; Ignas Dapšys

doi:10.3390/math9121344

A Comparison of Discrete Schemes for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

Mathematics ◽

10.3390/math9121344 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1344

Author(s):

Raimondas Čiegis ◽

Remigijus Čiegis ◽

Ignas Dapšys

Keyword(s):

Uniform Space ◽

Splitting Method ◽

Fractional Power ◽

Elliptic Operators ◽

Nonlocal Diffusion ◽

Parabolic Problems ◽

Test Problems ◽

Splitting Scheme ◽

The Stability ◽

Stability Results

The main aim of this article is to analyze the efficiency of general solvers for parabolic problems with fractional power elliptic operators. Such discrete schemes can be used in the cases of non-constant elliptic operators, non-uniform space meshes and general space domains. The stability results are proved for all algorithms and the accuracy of obtained approximations is estimated by solving well-known test problems. A modification of the second order splitting scheme is presented, it combines the splitting method to solve locally the nonlinear subproblem and the AAA algorithm to solve the nonlocal diffusion subproblem. Results of computational experiments are presented and analyzed.

Epitaxial Growth in Dislocation-Free Strained Alloy Films

MRS Proceedings ◽

10.1557/proc-715-a19.10 ◽

2002 ◽

Vol 715 ◽

Author(s):

Zhi-Feng Huang ◽

Rashmi C. Desai

Keyword(s):

Deposition Rate ◽

Elastic Moduli ◽

Composition Dependence ◽

Critical Thickness ◽

Lattice Misfit ◽

Misfit Strain ◽

Joint Stability ◽

Alloy Films ◽

The Stability ◽

Stability Results

AbstractThe morphological and compositional instabilities in the heteroepitaxial strained alloy films have attracted intense interest from both experimentalists and theorists. To understand the mechanisms and properties for the generation of instabilities, we have developed a nonequilibrium, continuum model for the dislocation-free and coherent film systems. The early evolution processes of surface pro.les for both growing and postdeposition (non-growing) thin alloy films are studied through a linear stability analysis. We consider the coupling between top surface of the film and the underlying bulk, as well as the combination and interplay of different elastic effects. These e.ects are caused by filmsubstrate lattice misfit, composition dependence of film lattice constant (compositional stress), and composition dependence of both Young's and shear elastic moduli. The interplay of these factors as well as the growth temperature and deposition rate leads to rich and complicated stability results. For both the growing.lm and non-growing alloy free surface, we determine the stability conditions and diagrams for the system. These show the joint stability or instability for film morphology and compositional pro.les, as well as the asymmetry between tensile and compressive layers. The kinetic critical thickness for the onset of instability during.lm growth is also calculated, and its scaling behavior with respect to misfit strain and deposition rate determined. Our results have implications for real alloy growth systems such as SiGe and InGaAs, which agree with qualitative trends seen in recent experimental observations.

Analysis of dengue model with fractal-fractional Caputo–Fabrizio operator

Advances in Difference Equations ◽

10.1186/s13662-020-02881-w ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Fatmawati ◽

Muhammad Altaf Khan ◽

Cicik Alfiniyah ◽

Ebraheem Alzahrani

Keyword(s):

Statistical Data ◽

Dengue Infection ◽

Fractional Operator ◽

Novel Approach ◽

Reduction Number ◽

Best Fitting ◽

The Stability ◽

Parameter Values ◽

Graphical Illustration ◽

Stability Results

AbstractIn this work, we study the dengue dynamics with fractal-factional Caputo–Fabrizio operator. We employ real statistical data of dengue infection cases of East Java, Indonesia, from 2018 and parameterize the dengue model. The estimated basic reduction number for this dataset is $\mathcal{R}_{0}\approx2.2020$ R 0 ≈ 2.2020 . We briefly show the stability results of the model for the case when the basic reproduction number is $\mathcal{R}_{0} <1$ R 0 < 1 . We apply the fractal-fractional operator in the framework of Caputo–Fabrizio to the model and present its numerical solution by using a novel approach. The parameter values estimated for the model are used to compare with fractal-fractional operator, and we suggest that the fractal-fractional operator provides the best fitting for real cases of dengue infection when varying the values of both operators’ orders. We suggest some more graphical illustration for the model variables with various orders of fractal and fractional.

Solving Vlasov-Poisson-Fokker-Planck Equations using NRxx method

Communications in Computational Physics ◽

10.4208/cicp.220415.080816a ◽

2017 ◽

Vol 21 (3) ◽

pp. 782-807 ◽

Cited By ~ 1

Author(s):

Yanli Wang ◽

Shudao Zhang

Keyword(s):

Splitting Method ◽

Linear Convergence ◽

Mass Conservation ◽

System A ◽

Spectral Convergence ◽

The Stability ◽

The Moment ◽

Stability And Accuracy ◽

Fokker Planck Equations ◽

Fokker Planck

AbstractWe present a numerical method to solve the Vlasov-Poisson-Fokker-Planck (VPFP) system using the NRxx method proposed in [4, 7, 9]. A globally hyperbolic moment system similar to that in [23] is derived. In this system, the Fokker-Planck (FP) operator term is reduced into the linear combination of the moment coefficients, which can be solved analytically under proper truncation. The non-splitting method, which can keep mass conservation and the balance law of the total momentum, is used to solve the whole system. A numerical problem for the VPFP system with an analytic solution is presented to indicate the spectral convergence with the moment number and the linear convergence with the grid size. Two more numerical experiments are tested to demonstrate the stability and accuracy of the NRxx method when applied to the VPFP system.

Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

Discrete and Continuous Dynamical Systems - Series S ◽

10.3934/dcdss.2021165 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Gregorio Díaz ◽

Jesús Ildefonso Díaz

Keyword(s):

Energy Balance ◽

Wiener Process ◽

Climate Models ◽

Random Parameter ◽

Parabolic Problems ◽

Change Of Variables ◽

Wiener Processes ◽

Cylindrical Wiener Process ◽

The Stability ◽

Suitable Change

<p style='text-indent:20px;'>We consider a class of one-dimensional nonlinear stochastic parabolic problems associated to Sellers and Budyko diffusive energy balance climate models with a Legendre weighted diffusion and an additive cylindrical Wiener processes forcing. Our results use in an important way that, under suitable assumptions on the Wiener processes, a suitable change of variables leads the problem to a pathwise random PDE, hence an essentially "deterministic" formulation depending on a random parameter. Two applications are also given: the stability of solutions when the Wiener process converges to zero and the asymptotic behaviour of solutions for large time.</p>

Boundary Stabilization of a Semilinear Wave Equation with Variable Coefficients under the Time-Varying and Nonlinear Feedback

Abstract and Applied Analysis ◽

10.1155/2014/728760 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Bei Gong ◽

Xiaopeng Zhao

Keyword(s):

Wave Equation ◽

Riemannian Geometry ◽

Nonlinear Term ◽

Variable Coefficients ◽

Time Varying ◽

Nonlinear Feedback ◽

Boundary Stabilization ◽

Semilinear Wave Equation ◽

The Stability ◽

Stability Results

We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.

On the probabilistic stability of some functional equations

Carpathian Journal of Mathematics ◽

10.37193/cjm.2013.01.01 ◽

2013 ◽

Vol 29 (1) ◽

pp. 125-132

Author(s):

CLAUDIA ZAHARIA ◽

◽

DOREL MIHET ◽

Keyword(s):

Fixed Point ◽

Functional Equations ◽

Fixed Point Theory ◽

Point Theory ◽

Quadratic Functional ◽

Normed Spaces ◽

Stability Of Functional Equations ◽

The Stability ◽

Stability Results

We establish stability results concerning the additive and quadratic functional equations in complete Menger ϕ-normed spaces by using fixed point theory. As particular cases, some theorems regarding the stability of functional equations in β - normed and quasi-normed spaces are obtained.

Travelling peakon and solitary wave solutions of modified Fornberg–Whitham equations with nonhomogeneous boundary conditions

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2021-0047 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

İhsan Çelikkaya

Keyword(s):

Solitary Wave ◽

Boundary Conditions ◽

Numerical Solutions ◽

Test Problems ◽

Von Neumann ◽

Nonhomogeneous Boundary ◽

Whitham Equations ◽

Fourier Series Method ◽

Nonhomogeneous Boundary Conditions ◽

The Stability

Abstract In this study, the numerical solutions of the modified Fornberg–Whitham (mFW) equation, which describes immigration of the solitary wave and peakon waves with discontinuous first derivative at the peak, have been obtained by the collocation finite element method using quintic trigonometric B-spline bases. Although there are solutions of this equation by semi-analytical and analytical methods in the literature, there are very few studies on the solution of the equation by numerical methods. Any linearization technique has not been used while applying the method. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. To show the performance of the method, we have considered three test problems with nonhomogeneous boundary conditions having analytical solutions. The error norms L 2 and L ∞ are calculated to demonstrate the accuracy and efficiency of the presented numerical scheme.

Proof of the stability results

On the Topology and Future Stability of the Universe ◽

10.1093/acprof:oso/9780199680290.003.0033 ◽

2013 ◽

pp. 620-633

Author(s):

Hans Ringström

Keyword(s):

The Stability ◽

Stability Results

On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls

Symmetry ◽

10.3390/sym11050712 ◽

2019 ◽

Vol 11 (5) ◽

pp. 712 ◽

Cited By ~ 2

Author(s):

Manuel De la Sen

Keyword(s):

Dynamic System ◽

Output Feedback ◽

Iteration Step ◽

Stability And Convergence ◽

Feedback Controls ◽

Time Aggregation ◽

The Stability ◽

The Given ◽

Stability Results

This paper links the celebrated Cauchy’s interlacing theorem of eigenvalues for partitioned updated sequences of Hermitian matrices with stability and convergence problems and results of related sequences of matrices. The results are also applied to sequences of factorizations of semidefinite matrices with their complex conjugates ones to obtain sufficiency-type stability results for the factors in those factorizations. Some extensions are given for parallel characterizations of convergent sequences of matrices. In both cases, the updated information has a Hermitian structure, in particular, a symmetric structure occurs if the involved vector and matrices are complex. These results rely on the relation of stable matrices and convergent matrices (those ones being intuitively stable in a discrete context). An epidemic model involving a clustering structure is discussed in light of the given results. Finally, an application is given for a discrete-time aggregation dynamic system where an aggregated subsystem is incorporated into the whole system at each iteration step. The whole aggregation system and the sequence of aggregated subsystems are assumed to be controlled via linear-output feedback. The characterization of the aggregation dynamic system linked to the updating dynamics through the iteration procedure implies that such a system is, generally, time-varying.

Stability Anomalies of Some Jacobian-Free Iterative Methods of High Order of Convergence

Axioms ◽

10.3390/axioms8020051 ◽

2019 ◽

Vol 8 (2) ◽

pp. 51

Author(s):

Alicia Cordero ◽

Javier G. Maimó ◽

Juan R. Torregrosa ◽

María P. Vassileva

Keyword(s):

Difference Operator ◽

Divided Difference ◽

Polynomial System ◽

Nonlinear Problems ◽

Coefficient Matrix ◽

Test Problems ◽

Stability Performance ◽

Need To Evaluate ◽

The Stability ◽

Theoretical Results

In this manuscript, we design two classes of parametric iterative schemes to solve nonlinear problems that do not need to evaluate Jacobian matrices and need to solve three linear systems per iteration with the same divided difference operator as the coefficient matrix. The stability performance of the classes is analyzed on a quadratic polynomial system, and it is shown that for many values of the parameter, only convergence to the roots of the problem exists. Finally, we check the performance of these methods on some test problems to confirm the theoretical results.