68 results
Search Results
2. New escape conditions with general complex polynomial for fractals via new fixed point iteration
- Author
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Yu-Pei Lv, Sumaira Nawaz, Muhammad Tanveer, Ali Raza, and Imran Ahmed
- Subjects
General Mathematics ,lcsh:Mathematics ,State (functional analysis) ,Fixed point ,Mandelbrot set ,lcsh:QA1-939 ,mandelbrot set ,Fractal ,Quadratic equation ,fractal ,fixed point ,Fixed-point iteration ,Scheme (mathematics) ,general polynomial ,Applied mathematics ,Orbit (control theory) ,Mathematics ,multi-corns set - Abstract
The aim of this paper is to generalize the results regarding fractals and prove escape conditions for general complex polynomial. In this paper we state the orbit of a newly defined iterative scheme and establish the escape criteria in fractal generation for general complex polynomial. We use established escape criteria in algorithms to generate Mandelbrot and Multi-corns sets. In addition, we present some graphs of quadratic, cubic and higher Mandelbrot and Multi-corns sets and discuss how the alteration in parameters make changes in graphs.
- Published
- 2021
3. Oscillation theorems for higher order dynamic equations with superlinear neutral term
- Author
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Jehad Alzabut, Kamaleldin Abodayeh, and Said R. Grace
- Subjects
Class (set theory) ,Oscillation ,General Mathematics ,lcsh:Mathematics ,Applied mathematics ,Order (group theory) ,oscillation criteria ,higher order dynamic equations ,lcsh:QA1-939 ,Dynamic equation ,superlinear neutral term ,Term (time) ,Mathematics - Abstract
In this paper, several oscillation criteria for a class of higher order dynamic equations with superlinear neutral term are established. The proposed results provide a unified platform that adequately covers both discrete and continuous equations and further sufficiently comments on oscillatory behavior of more general class of equations than the ones reported in the literature. We conclude the paper by demonstrating illustrative examples.
- Published
- 2021
4. Numerical simulation of the fractal-fractional reaction diffusion equations with general nonlinear
- Author
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Manal Alqhtani and Khaled M. Saad
- Subjects
Computer simulation ,Differential equation ,lagrange polynomial interpolation ,General Mathematics ,lcsh:Mathematics ,the fractal-fractional reaction diffusion equations ,lcsh:QA1-939 ,Fractal dimension ,Nonlinear system ,the exponential law ,Fractal ,Kernel (statistics) ,Reaction–diffusion system ,the power law ,Applied mathematics ,Exponential decay ,generalized mittag-leffler function ,Mathematics - Abstract
In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, 1
- Published
- 2021
5. The new reflected power function distribution: Theory, simulation & application
- Author
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Riffat Jabeen, Ahmad Saeed Akhter, and Azam Zaka
- Subjects
Percentile ,Distribution (number theory) ,reflected power function distribution ,General Mathematics ,lcsh:Mathematics ,Order statistic ,Truncated mean ,Estimator ,power function distribution ,Function (mathematics) ,percentile estimator ,lcsh:QA1-939 ,characterization of truncated distribution ,Applied mathematics ,Applied science ,Power function ,Mathematics - Abstract
The aim of the paper is to propose a new Reflected Power function distribution (RPFD). We provide the various properties of the new model in detail such as moments, vitality function and order statistics. We characterize the RPFD based on conditional moments (Right and Left Truncated mean) and doubly truncated mean. We also study the shape of the new distribution to be applicable in many real life situations. We estimate the parameters for the proposed RPFD by using different methods such as maximum likelihood method, modified maximum likelihood method, percentile estimator and modified percentile estimator. The aim of the study is to increase the application of the Power function distribution (PFD). Using two different data sets from real life, we conclude that the RPFD perform better as compare to different competitor models already exist in the literature. We hope that the findings of this paper will be useful for researchers in different field of applied sciences.
- Published
- 2020
6. The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order
- Author
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Lei Hu and Shuqin Zhang
- Subjects
Lyapunov function ,Differential equation ,General Mathematics ,lcsh:Mathematics ,existence ,derivatives and integrals of variable order ,lcsh:QA1-939 ,differential equations of variable order ,piecewise constant functions ,symbols.namesake ,Nonlinear system ,Schauder fixed point theorem ,generalized lyapunov-type inequality ,symbols ,Piecewise ,Applied mathematics ,Boundary value problem ,Constant function ,Mathematics ,Variable (mathematics) - Abstract
In this paper, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.
- Published
- 2020
7. Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type
- Author
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Yilin Wang, Yibing Sun, Yige Zhao, and Zhi Liu
- Subjects
General Mathematics ,lcsh:Mathematics ,existence ,Existence theorem ,Fixed-point theorem ,Type (model theory) ,Expression (computer science) ,Differential operator ,Lipschitz continuity ,lcsh:QA1-939 ,mixed perturbations ,Banach algebra ,boundary value problem ,fractional differential equation ,Applied mathematics ,Boundary value problem ,Mathematics - Abstract
In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Caratheodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.
- Published
- 2020
8. New iterative approach for the solutions of fractional order inhomogeneous partial differential equations
- Author
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Rashid Nawaz, Sumbal Ahsan, Kottakkaran Sooppy Nisar, Dumitru Baleanu, and Laiq Zada
- Subjects
Partial differential equation ,Laplace transform ,Iterative method ,General Mathematics ,lcsh:Mathematics ,fractional order inhomogeneous system ,Interval (mathematics) ,fractional calculus ,lcsh:QA1-939 ,approximate solutions ,Fractional calculus ,Transformation (function) ,Integer ,fractional order roseau-hyman equation ,Applied mathematics ,Decomposition method (constraint satisfaction) ,new iterative method ,Mathematics - Abstract
In this paper, the study of fractional order partial differential equations is made by using the reliable algorithm of the new iterative method (NIM). The fractional derivatives are considered in the Caputo sense whose order belongs to the closed interval [0, 1]. The proposed method is directly extended to study the fractional-order Roseau-Hyman and fractional order inhomogeneous partial differential equations without any transformation to convert the given problem into integer order. The obtained results are compared with those obtained by Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Laplace Variational Iteration Method (LVIM) and the Laplace Adominan Decomposition Method (LADM). The results obtained by NIM, show higher accuracy than HPM, LVIM and LADM. The accuracy of the proposed method improves by taking more iterations.
- Published
- 2021
9. Error bounds for generalized vector inverse quasi-variational inequality Problems with point to set mappings
- Author
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J. F. Tang, X. R. Wang, M. Liu, S. S. Chang, and Salahuddin
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residual gap function ,General Mathematics ,lcsh:Mathematics ,Hausdorff space ,Solution set ,Inverse ,hausdorff lipschitz continuity ,Monotonic function ,Function (mathematics) ,error bounds ,Lipschitz continuity ,Residual ,relaxed monotonicity ,lcsh:QA1-939 ,generalized f-projection operator ,regularized gap function ,Variational inequality ,Applied mathematics ,generalized vector inverse quasi-variational inequality problems ,global gap function ,bi-mapping ,Mathematics ,strong monotonicity - Abstract
The goal of this paper is further to study a kind of generalized vector inverse quasi-variational inequality problems and to obtain error bounds in terms of the residual gap function, the regularized gap function, and the global gap function by utilizing the relaxed monotonicity and Hausdorff Lipschitz continuity. These error bounds provide effective estimated distances between an arbitrary feasible point and the solution set of generalized vector inverse quasi-variational inequality problems.
- Published
- 2021
10. The stationary distribution of a stochastic rumor spreading model
- Author
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Dapeng Gao, Peng Guo, and Chaodong Chen
- Subjects
Lyapunov function ,Stationary distribution ,Stochastic modelling ,General Mathematics ,lcsh:Mathematics ,White noise ,Rumor ,lcsh:QA1-939 ,stationary distribution ,symbols.namesake ,rumor spreading ,symbols ,threshold ,Applied mathematics ,Ergodic theory ,Uniqueness ,Persistence (discontinuity) ,Mathematics - Abstract
In this paper, we develop a rumor spreading model by introducing white noise into the model. We establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the stochastic model by constructing a suitable stochastic Lyapunov function, which provides us a good description of persistence. Finally, we provide some numerical simulations to illustrate the analytical results.
- Published
- 2021
11. Finite element approximation of time fractional optimal control problem with integral state constraint
- Author
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Jie Liu and Zhaojie Zhou
- Subjects
Discretization ,General Mathematics ,lcsh:Mathematics ,a priori error estimate ,space time finite element method ,Optimal control ,lcsh:QA1-939 ,integral state constraint ,Finite element method ,Piecewise linear function ,Scheme (mathematics) ,Piecewise ,A priori and a posteriori ,Applied mathematics ,time fractional optimal control problem ,Constant (mathematics) ,Mathematics - Abstract
In this paper we investigate the finite element approximation of time fractional optimal control problem with integral state constraint. A space-time finite element scheme for the control problem is developed with piecewise constant time discretization and piecewise linear spatial discretization for the state equation. A priori error estimate for the space-time discrete scheme is derived. Projected gradient algorithm is used to solve the discrete optimal control problem. Numerical experiments are carried out to illustrate the theoretical findings.
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- 2021
12. A relaxed generalized Newton iteration method for generalized absolute value equations
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Senlai Zhu, Yang Cao, and Shi Quan
- Subjects
Generalized Jacobian ,Iterative method ,General Mathematics ,lcsh:Mathematics ,Positive-definite matrix ,globally convergence ,lcsh:QA1-939 ,symbols.namesake ,generalized absolute value equations ,relaxation ,Fixed-point iteration ,Absolute value equation ,symbols ,newton method ,Applied mathematics ,Well-defined ,Coefficient matrix ,Newton's method ,Mathematics - Abstract
To avoid singular generalized Jacobian matrix and further accelerate the convergence of the generalized Newton (GN) iteration method for solving generalized absolute value equations Ax - B|x| = b, in this paper we propose a new relaxed generalized Newton (RGN) iteration method by introducing a relaxation iteration parameter. The new RGN iteration method involves the well-known GN iteration method and the Picard iteration method as special cases. Theoretical analyses show that the RGN iteration method is well defined and globally linearly convergent under suitable conditions. In addition, a specific sufficient condition is studied when the coefficient matrix A is symmetric positive definite. Finally, two numerical experiments arising from the linear complementarity problems are used to illustrate the effectiveness of the new RGN iteration method.
- Published
- 2021
13. A delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response
- Author
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Anwar Zeb, Ranjit Kumar Upadhyay, A. Pratap, and Yougang Wang
- Subjects
Lyapunov function ,Hopf bifurcation ,delay ,General Mathematics ,Addiction ,media_common.quotation_subject ,lcsh:Mathematics ,Functional response ,periodic solution ,stability ,lcsh:QA1-939 ,Two stages ,Critical point (mathematics) ,Synthetic drugs ,symbols.namesake ,symbols ,Applied mathematics ,synthetic drugs model ,hopf bifurcation ,Bifurcation ,media_common ,Mathematics - Abstract
This paper gropes the stability and Hopf bifurcation of a delayed synthetic drug transmission model with two stages of addiction and Holling Type-II functional response. The critical point at which a Hopf bifurcation occurs can be figured out by using the escalating time delay of psychologically addicts as a bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are explored with aid of the central manifold theorem and normal form theory. Specially, global stability of the model is proved by constructing a suitable Lyapunov function. To underline effectiveness of the obtained results and analyze influence of some influential parameters on dynamics of the model, some numerical simulations are ultimately addressed.
- Published
- 2021
14. Stability of general pathogen dynamic models with two types of infectious transmission with immune impairment
- Author
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B. S. Alofi and S. A. Azoz
- Subjects
Lyapunov function ,pathogen infection ,Steady state (electronics) ,General Mathematics ,lcsh:Mathematics ,cell-to-cell transmission ,lcsh:QA1-939 ,Stability (probability) ,global stability ,Quantitative Biology::Cell Behavior ,symbols.namesake ,immune impairment ,Transmission (telecommunications) ,Exponential stability ,Stability theory ,Bounded function ,symbols ,Applied mathematics ,Quantitative Biology::Populations and Evolution ,Basic reproduction number ,Mathematics - Abstract
In this paper, we investigate the global properties of two general models of pathogen infection with immune deficiency. Both pathogen-to-cell and cell-to-cell transmissions are considered. Latently infected cells are included in the second model. We show that the solutions are nonnegative and bounded. Lyapunov functions are organized to prove the global asymptotic stability for uninfected and infected steady states of the models. Analytical expressions for the basic reproduction number $\mathcal{R}_{0}$ and the necessary condition under which the uninfected and infected steady states are globally asymptotically stable are established. We prove that if $\mathcal{R}_{0}$ < 1 then the uninfected steady state is globally asymptotically stable (GAS), and if $\mathcal{R}_{0}$ > 1 then the infected steady state is GAS. Numerical simulations are performed and used to support the analytical results.
- Published
- 2021
15. On the nonstandard numerical discretization of SIR epidemic model with a saturated incidence rate and vaccination
- Author
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Isnani Darti and Agus Suryanto
- Subjects
Lyapunov function ,Discretization ,Continuous modelling ,General Mathematics ,lcsh:Mathematics ,Finite difference ,dynamically-consistent discretization ,Function (mathematics) ,Nonstandard finite difference scheme ,saturated incidence rate ,local and global stability analysis ,lcsh:QA1-939 ,Euler method ,symbols.namesake ,symbols ,Applied mathematics ,sir epidemic model ,Epidemic model ,lyapunov function ,Mathematics - Abstract
Recently, Hoang and Egbelowo (Boletin de la Sociedad Matematica Mexicana, 2020) proposed a nonstandard finite difference scheme (NSFD) to get a discrete SIR epidemic model with saturated incidence rate and constant vaccination. The discrete model was derived by discretizing the right-hand sides of the system locally and the first order derivative is approximated by the generalized forward difference method but with a restrictive denominator function. Their analysis showed that the NSFD scheme is dynamically-consistent only for relatively small time-step sizes. In this paper, we propose and analyze an alternative NSFD scheme by applying nonlocal approximation and choosing the denominator function such that the proposed scheme preserves the boundedness of solutions. It is verified that the proposed discrete model is dynamically-consistent with the corresponding continuous model for all time-step size. The analytical results have been confirmed by some numerical simulations. We also show numerically that the proposed NSFD scheme is superior to the Euler method and the NSFD method proposed by Hoang and Egbelowo (2020).
- Published
- 2021
16. Determination of three parameters in a time-space fractional diffusion equation
- Author
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Xiang-Tuan Xiong, Xuemin Xue, and Wanxia Shi
- Subjects
Nonlinear inverse problem ,fractional diffusion equation ,General Mathematics ,lcsh:Mathematics ,uniqueness ,lcsh:QA1-939 ,Regularization (mathematics) ,Tikhonov regularization ,regularization ,Time space ,Boundary data ,Fractional diffusion ,Applied mathematics ,Uniqueness ,Diffusion (business) ,ill-posedness ,Mathematics - Abstract
In this paper, we consider a nonlinear inverse problem of recovering two fractional orders and a diffusion coefficient in a one-dimensional time-space fractional diffusion equation. The uniqueness of fractional orders and the diffusion coefficient, characterizing slow diffusion, can be obtained from the accessible boundary data. Two computational methods, Tikhonov method and Levenberg-Marquardt method, are proposed to solving this problem. Finally, an example is presented to illustrate the efficiency of the two numerical algorithm.
- Published
- 2021
17. The analytical analysis of nonlinear fractional-order dynamical models
- Author
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Jiabin Xu, Dumitru Baleanu, Rasool Shah, Hassan Khan, Shaban Aly, and A.A. Alderremy
- Subjects
Laplace transform ,General Mathematics ,Homotopy ,lcsh:Mathematics ,Boundary (topology) ,lcsh:QA1-939 ,Fractional calculus ,Nonlinear system ,swift-hohenberg equation ,Bounded function ,caputo operator ,Fluid dynamics ,Applied mathematics ,laplace transform ,adomian decomposition method ,Adomian decomposition method ,Mathematics - Abstract
The present research paper is related to the analytical solution of fractional-order nonlinear Swift-Hohenberg equations using an efficient technique. The presented model is related to the temperature and thermal convection of fluid dynamics which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In this work Laplace Adomian decomposition method is implemented because it require small volume of calculations. Unlike the variational iteration method and Homotopy pertubation method, the suggested technique required no variational parameter and having simple calculation of fractional derivative respectively. Numerical examples verify the validity of the suggested method. It is confirmed that the present method's solutions are in close contact with the solutions of other existing methods. It is also investigated through graphs and tables that the suggested method's solutions are almost identical with different analytical methods.
- Published
- 2021
18. Fast Crank-Nicolson compact difference scheme for the two-dimensional time-fractional mobile/immobile transport equation
- Author
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Congcong Li, Lijuan Nong, An Chen, and Qian Yi
- Subjects
fast discrete sine transform ,non-smooth solution ,General Mathematics ,lcsh:Mathematics ,mobile/immobile transport equation ,Order (ring theory) ,lcsh:QA1-939 ,compact difference operator ,Alpha (programming language) ,Operator (computer programming) ,Discrete sine transform ,Scheme (mathematics) ,Applied mathematics ,Crank–Nicolson method ,modified l1 method ,Convection–diffusion equation ,Laplace operator ,Mathematics - Abstract
In this paper, we consider the efficient numerical scheme for solving time-fractional mobile/immobile transport equation. By utilizing the compact difference operator to approximate the Laplacian, we develop an efficient Crank-Nicolson compact difference scheme based on the modified L1 method. It is proved that the proposed scheme is stable with the accuracy of $ O(\tau^{2-\alpha}+h^4) $, where $ \tau $ and $ h $ are respectively the temporal and spatial stepsizes, and the fractional order $ \alpha\in(0, 1) $. In addition, we improve the computational performance for the non-smooth issue by the fast discrete sine transform technology and the method of adding correction terms. Finally, numerical examples are provided to verify the effectiveness of the proposed scheme.
- Published
- 2021
19. Strong Langmuir turbulence dynamics through the trigonometric quintic and exponential B-spline schemes
- Author
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Mostafa M. A. Khater and A. El-Sayed Ahmed
- Subjects
Computer simulation ,Langmuir Turbulence ,General Mathematics ,B-spline ,lcsh:Mathematics ,the trigonometric quintic (tqbs) and exponential b-spline (ecbs) schemes ,nonlinear klein-gordon-zakharov (kgz) model ,lcsh:QA1-939 ,Quintic function ,Exponential function ,Nonlinear system ,numerical simulation ,Applied mathematics ,Boundary value problem ,Adomian decomposition method ,Mathematics - Abstract
In this manuscript, two recent numerical schemes (the trigonometric quintic and exponential cubic B-spline schemes) are employed for evaluating the approximate solutions of the nonlinear Klein-Gordon-Zakharov model. This model describes the interaction between the Langmuir wave and the ion-acoustic wave in a high-frequency plasma. The initial and boundary conditions are constructed via a novel general computational scheme. [ 1 ] has used five different numerical schemes, such as the Adomian decomposition method, Elkalla-expansion method, three-member of the well-known cubic B-spline schemes. Consequently, the comparison between our solutions and that have been given in [ 1 ], shows the accuracy of seven recent numerical schemes along with the considered model. The obtained numerical solutions are sketched in two dimensional and column distribution to explain the matching between the computational and numerical simulation. The novelty, originality, and accuracy of this research paper are explained by comparing the obtained numerical solutions with the previously obtained solutions.
- Published
- 2021
20. New generalizations for Gronwall type inequalities involving a ψ-fractional operator and their applications
- Author
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Jehad Alzabut, Mutti-Ur Rehman, Yassine Adjabi, and Weerawat Sudsutad
- Subjects
generalized gronwall's inequality ,Inequality ,Differential equation ,ulam-hyers stability ,General Mathematics ,media_common.quotation_subject ,lcsh:Mathematics ,Stability (learning theory) ,Type (model theory) ,lcsh:QA1-939 ,Fractional operator ,Operator (computer programming) ,Initial value problem ,Applied mathematics ,Uniqueness ,ψ-fractional operators ,ψ-fractional initial value problem ,Mathematics ,media_common ,existence and uniqueness - Abstract
In this paper, we provide new generalizations for the Gronwall's inequality in terms of a $ \psi $-fractional operator. The new forms of Gronwall's inequality are obtained within a general platform that includes several existing results as particular cases. To apply our results and examine their validity, we prove the existence and uniqueness of solutions for $ \psi $-fractional initial value problem. Further, the Ulam-Hyers stability of solutions for $ \psi $-fractional differential equations is discussed. For the sake of illustrating the proposed results, we give some particular examples.
- Published
- 2021
21. Square-mean asymptotically almost periodic solutions of second order nonautonomous stochastic evolution equations
- Author
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Litao Zhang and Jinghuai Liu
- Subjects
Class (set theory) ,General Mathematics ,lcsh:Mathematics ,Hilbert space ,second order nonautonomous stochastic evolution equations ,Stochastic evolution ,lcsh:QA1-939 ,Square (algebra) ,symbols.namesake ,symbols ,mild solution ,Order (group theory) ,Applied mathematics ,Uniqueness ,square-mean asymptotically almost periodic function ,Mathematics - Abstract
In this paper, we study the existence of square-mean asymptotically almost periodic mild solutions for a class of second order nonautonomous stochastic evolution equations in Hilbert spaces. By using the principle of Banach contractive mapping principle, the existence and uniqueness of square-mean asymptotically almost periodic mild solutions of the equation are obtained. To illustrate the abstract result, a concrete example is given.
- Published
- 2021
22. On finite-time stability and stabilization of nonlinear hybrid dynamical systems
- Author
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Wassim M. Haddad and Junsoo Lee
- Subjects
Lyapunov function ,Dynamical systems theory ,General Mathematics ,lcsh:Mathematics ,impulsive systems ,Scalar (physics) ,hybrid control ,finite time stability ,Lipschitz continuity ,lcsh:QA1-939 ,Stability (probability) ,System dynamics ,symbols.namesake ,Nonlinear system ,Hybrid system ,symbols ,Applied mathematics ,finite time stabilization ,Mathematics - Abstract
Finite time stability involving dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time have been studied for both continuous-time and discrete-time systems. For continuous-time systems, finite time stability is defined for equilibria of continuous but non-Lipschitzian nonlinear dynamics, whereas discrete-time systems can exhibit finite time stability even when the system dynamics are linear, and hence, Lipschitz continuous. Alternatively, for impulsive dynamical systems it may be possible to reset the system states to an equilibrium state achieving finite time stability without requiring a non-Lipschitz condition for the continuous-time part of the hybrid system dynamics. In this paper, we develop sufficient Lyapunov conditions for finite time stability of impulsive dynamical systems using both a scalar differential Lyapunov inequality on the continuous-time dynamics as well as a scalar difference Lyapunov inequality on the discrete-time resetting dynamics. Furthermore, using our proposed finite time stability results, we design universal hybrid finite time stabilizing control laws for impulsive dynamical systems. Finally, we present several numerical examples for finite time stabilization of network impulsive dynamical systems.
- Published
- 2021
23. Ulam stabilities of nonlinear coupled system of fractional differential equations including generalized Caputo fractional derivative
- Author
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Tamer Nabil
- Subjects
ψ-caputo operator ,General Mathematics ,lcsh:Mathematics ,coupled implicit system ,Fixed point ,lcsh:QA1-939 ,Fractional operator ,Stability (probability) ,Fractional calculus ,Nonlinear system ,Applied mathematics ,fixed-point ,Uniqueness ,Fractional differential ,ulam stability ,existence of solution ,Mathematics - Abstract
In this paper, we establish the existence and uniqueness of solution for a nonlinear coupled system of implicit fractional differential equations including $ \psi $-Caputo fractional operator under nonlocal conditions. Schaefer's and Banach fixed-point theorems are applied to obtain the solvability results for the proposed system. Furthermore, we extend the results to investigate several types of Ulam stability for the proposed system by using classical tool of nonlinear analysis. Finally, an example is provided to illustrate the abstract results.
- Published
- 2021
24. Endpoint estimates for multilinear fractional singular integral operators on Herz and Herz type Hardy spaces
- Author
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Dazhao Chen
- Subjects
Multilinear map ,Pure mathematics ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,herz space ,Type (model theory) ,Space (mathematics) ,Lebesgue integration ,symbols.namesake ,Operator (computer programming) ,multilinear operator ,Singular integral operators ,Mathematics ,Mathematics::Functional Analysis ,herz type hardy space ,Mathematics::Operator Algebras ,Applied Mathematics ,lcsh:Mathematics ,General Medicine ,Hardy space ,lcsh:QA1-939 ,fractional singular integral operators ,Computational Mathematics ,Modeling and Simulation ,symbols ,General Agricultural and Biological Sciences ,bmo space - Abstract
The boundedness of singular and fractional integral operator on Lebesgue and Hardy spaces have been well studied. The theory of Herz space and Herz type Hardy space, as a local version of Lebesgue and Hardy space, have been developed. The main purpose of this paper is to establish the endpoint continuity properties of some multilinear operators related to certain non-convolution type fractional singular integral operators on Herz and Herz type Hardy spaces and the endpoint estimates for the multilinear operators on Herz and Herz type Hardy spaces are obtained.
- Published
- 2021
25. Numerical method for pricing discretely monitored double barrier option by orthogonal projection method
- Author
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Kazem Nouri, Dumitru Baleanu, Leila Torkzadeh, and Milad Fahimi
- Subjects
Chebyshev polynomials ,Partial differential equation ,General Mathematics ,Numerical analysis ,lcsh:Mathematics ,Orthographic projection ,orthogonal projection ,double barrier option ,chebyshev polynomial ,black-scholes model ,Black–Scholes model ,lcsh:QA1-939 ,Projection method ,Benchmark (computing) ,Applied mathematics ,Heat equation ,Mathematics - Abstract
In this paper, we consider discretely monitored double barrier option based on the Black-Scholes partial differential equation. In this scenario, the option price can be computed recursively upon the heat equation solution. Thus we propose a numerical solution by projection method. We implement this method by considering the Chebyshev polynomials of the second kind. Finally, numerical examples are carried out to show accuracy of the presented method and demonstrate acceptable accordance of our method with other benchmark methods.
- Published
- 2021
26. Extremal solutions of φ−Caputo fractional evolution equations involving integral kernels
- Author
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Parinya Sa Ngiamsunthorn and Apassara Suechoei
- Subjects
Volterra operator ,Semigroup ,General Mathematics ,Fredholm operator ,volterra operator ,lcsh:Mathematics ,fredholm operator ,upper and lower solutions ,lcsh:QA1-939 ,Fractional calculus ,fractional evolution equation ,Nonlinear system ,Monotone polygon ,Applied mathematics ,Initial value problem ,monotone iterative technique ,Uniqueness ,Mathematics - Abstract
This paper deals with the existence and uniqueness of solution for the Cauchy problem of $ \varphi- $Caputo fractional evolution equations involving Volterra and Fredholm integral kernels. We derive a mild solution in terms of semigroup and construct a monotone iterative sequence for extremal solutions under a noncompactness measure condition of the nonlinearity. These results can be reduced to previous works with the classical Caputo fractional derivative. Furthermore, we give an example of initial-boundary value problem for the time-fractional parabolic equation to illustrate the application of the results.
- Published
- 2021
27. Computational study of the convection-diffusion equation using new cubic B-spline approximations
- Author
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Muhammad Yaseen, Aatika Yousaf, Rekha Srivastava, and Asifa Tassaddiq
- Subjects
General Mathematics ,B-spline ,lcsh:Mathematics ,Stability (learning theory) ,Finite difference ,Derivative ,stability ,lcsh:QA1-939 ,new cubic b-spline approximations ,Position (vector) ,Time derivative ,Convergence (routing) ,convection-diffusion equation ,Applied mathematics ,cubic b-splines ,Convection–diffusion equation ,Mathematics - Abstract
This paper introduces an efficient numerical procedure based on cubic B-Spline (CuBS) with a new approximation for the second-order space derivative for computational treatment of the convection-diffusion equation (CDE). The time derivative is approximated using typical finite differences. The key benefit of the scheme is that the numerical solution is obtained as a smooth piecewise continuous function which empowers one to find approximate solution at any desired position in the domain. Further, the new approximation has considerably increased the accuracy of the scheme. A stability analysis is performed to assure that the errors do not magnify. Convergence analysis of the scheme is also discussed. The scheme is implemented on some test problems and the outcomes are contrasted with those of some current approximating techniques from the literature. It is concluded that the offered scheme is equitably superior and effective.
- Published
- 2021
28. On a combination of fractional differential and integral operators associated with a class of normalized functions
- Author
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Rabha W. Ibrahim and Dumitru Baleanu
- Subjects
Class (set theory) ,Differential equation ,General Mathematics ,lcsh:Mathematics ,fractional calculus ,univalent function ,lcsh:QA1-939 ,Unit disk ,briot-bouquet differential equation ,Domain (mathematical analysis) ,analytic function ,Fractional calculus ,Computer Science::Other ,fractional differential operator ,Applied mathematics ,subordination and superordination ,Convex function ,open unit disk ,Univalent function ,Mathematics ,Analytic function - Abstract
Recently, the combined fractional operator (CFO) is introduced and discussed in Baleanu et al. [ 1 ] in real domain. In this paper, we extend CFO to the complex domain and study its geometric properties in some normalized analytic functions including the starlike and convex functions. Moreover, we employ the complex CFO to modify a class of Briot-Bouquet differential equations in a complex region. As a consequence, the upper solution is illustrated by using the concept of subordination inequality.
- Published
- 2021
29. New stability criteria for semi-Markov jump linear systems with time-varying delays
- Author
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Wentao Le, Yucai Ding, Wenqing Wu, and Hui Liu
- Subjects
Stochastic stability ,General Mathematics ,lcsh:Mathematics ,wirtinger's integral inequality ,time-varying delays ,Linear matrix ,lcsh:QA1-939 ,Stability (probability) ,semi-markov jump linear systems ,Markov jump linear systems ,Markovian jump linear systems ,delay-dependent stability ,Computer Science::Systems and Control ,Applied mathematics ,Lyapunov matrix ,Mathematics - Abstract
In this paper, the delay-dependent stochastic stability problem of semi-Markov jump linear systems (S-MJLS) with time-varying delays is investigated. By constructing a Lyapunov-Krasovskii functional (LKF) with two delay-product-type terms, a new sufficient condition on stochastic stability of S-MJLSs is derived in terms of linear matrix inequalities (LMIs). Furthermore, the combination use of a slack condition on Lyapunov matrix and the improved Wirtinger's integral inequality reduces the conservatism of the result. Numerical examples are provided to verify the effectiveness and superiority of the presented results.
- Published
- 2021
30. On implicit coupled systems of fuzzy fractional delay differential equations with triangular fuzzy functions
- Author
-
Heng-you Lan, Chang-jiang Liu, and Yu-ting Wu
- Subjects
Work (thermodynamics) ,Mathematics::General Mathematics ,General Mathematics ,lcsh:Mathematics ,fuzzy terms separation method ,Delay differential equation ,Function (mathematics) ,Type (model theory) ,lcsh:QA1-939 ,Fuzzy logic ,Matrix (mathematics) ,solution algorithm ,Applied mathematics ,Uniqueness ,triangular fuzzy function ,implicit coupled system ,fuzzy fractional delay differential equation ,Mathematics ,existence and uniqueness - Abstract
In this paper, we introduce and study an implicit coupled system of fuzzy fractional delay differential equations involving fuzzy initial values and fuzzy source functions of triangular type. We assume that these initial values and source functions are triangular fuzzy functions and define the solution of the implicit coupled system as a triangular fuzzy function matrix consisting of real functional matrices. The method of triangular fuzzy function, fractional steps and fuzzy terms separation are used to solve the implicit coupled systems. Further, we prove existence and uniqueness of solution for the considered systems, and also construct a solution algorithm. Finally, an example is given to illustrate our main results and some further work are presented.
- Published
- 2021
31. Some notes on possibilistic variances of generalized trapezoidal intuitionistic fuzzy numbers
- Author
-
Qiansheng Zhang and Dongfang Sun
- Subjects
generalized trapezoidal intuitionistic fuzzy number ,Uncertain data ,Generalization ,possibilistic covariance ,General Mathematics ,Decision theory ,lcsh:Mathematics ,Intuitionistic fuzzy ,Variance (accounting) ,lcsh:QA1-939 ,Probability theory ,possibilistic variance ,Applied mathematics ,Fuzzy number ,possibilistic mean ,Mathematics - Abstract
Intuitionistic fuzzy number as the generalization of fuzzy number has better capability to model uncertain data in management decision science and mathematical engineering problem. In this paper, we present a new definition of possibilistic mean and variance for generalized trapezoidal intuitionistic fuzzy number (GTIFN), and show that some properties of crisp mean and variance in probability theory are preserved by possibilistic mean and variance of generalized trapezoidal intuitionistic fuzzy numbers. We also discuss some notes on possibilistic variance of GTIFNs and get the important relationships between the proposed two possibilistic variances of generalized trapezoidal intuitionistic fuzzy number.
- Published
- 2021
32. Numerical simulations for initial value inversion problem in a two-dimensional degenerate parabolic equation
- Author
-
Zui-Cha Deng, Liu Yang, and Fan-Li Liu
- Subjects
Finite volume method ,Computer simulation ,Iterative method ,General Mathematics ,lcsh:Mathematics ,Degenerate energy levels ,MathematicsofComputing_NUMERICALANALYSIS ,difference scheme ,Inverse problem ,stability ,lcsh:QA1-939 ,Landweber iteration ,degenerate parabolic equation ,Conjugate gradient method ,numerical simulation ,Applied mathematics ,Initial value problem ,initial value inversion problem ,Mathematics - Abstract
In this paper, we study the inverse problem of identifying the initial value of a two-dimensional degenerate parabolic equation, which often appears in the fields of engineering, physics, and computer image processing. Firstly, the difference scheme of forward problem is established by using the finite volume method. Then stability and convergence of the difference equations are proved rigorously. Finally, the Landweber iteration and conjugate gradient method are used to solve the inverse problem, and some typical numerical examples are shown to verify the validity of our iterative algorithm. Numerical results show that the algorithm is stable and efficient.
- Published
- 2021
33. Some generalized fractional integral inequalities with nonsingular function as a kernel
- Author
-
Shahid Mubeen, Iqra Nayab, Dumitru Baleanu, Rana Safdar Ali, Kottakkaran Sooppy Nisar, and Gauhar Rahman
- Subjects
convexity ,General Mathematics ,lcsh:Mathematics ,Function (mathematics) ,Type (model theory) ,lcsh:QA1-939 ,Convexity ,law.invention ,inequalities and integral operators ,symbols.namesake ,Invertible matrix ,Operator (computer programming) ,law ,Hadamard transform ,Kernel (statistics) ,symbols ,Applied mathematics ,generalized multi-index bessel function ,fractional derivatives and integrals ,Bessel function ,Mathematics - Abstract
Integral inequalities play a key role in applied and theoretical mathematics. The purpose of inequalities is to develop mathematical techniques in analysis. The goal of this paper is to develop a fractional integral operator having a non-singular function (generalized multi-index Bessel function) as a kernel and then to obtain some significant inequalities like Hermit Hadamard Mercer inequality, exponentially $ (s-m) $-preinvex inequalities, Polya-Szego and Chebyshev type integral inequalities with the newly developed fractional operator. These results describe in general behave and provide the extension of fractional operator theory (FOT) in inequalities.
- Published
- 2021
34. Transportation inequalities for doubly perturbed stochastic differential equations with Markovian switching
- Author
-
Liping Xu, Jie Zhou, Zhi Li, and Weiguo Liu
- Subjects
Class (set theory) ,Girsanov theorem ,General Mathematics ,lcsh:Mathematics ,transportation inequalities ,lcsh:QA1-939 ,Stochastic differential equation ,girsanov transformation ,Quadratic equation ,Transformation (function) ,Metric (mathematics) ,Applied mathematics ,Markovian switching ,Focus (optics) ,doubly perturbed sdes ,markovian switching ,Mathematics - Abstract
In this paper, we focus on a class of doubly perturbed stochastic differential equations with Markovian switching. Using the Girsanov transformation argument we establish the quadratic transportation inequalities for the law of the solution of those equations with Markovian switching under the $ d_2 $ metric and the uniform metric $ d_{\infty} $.
- Published
- 2021
35. Set-valued variational inclusion problem with fuzzy mappings involving XOR-operation
- Author
-
Javid Iqbal, Imran Ali, Puneet Kumar Arora, and Waseem Ali Mir
- Subjects
Fuzzy mapping ,algorithm ,Iterative method ,General Mathematics ,fuzzy mapping ,lcsh:Mathematics ,lcsh:QA1-939 ,Fuzzy logic ,Set (abstract data type) ,operation ,inclusion ,Convergence (routing) ,Resolvent operator ,Applied mathematics ,set-valued ,Approximate solution ,Bitwise operation ,Mathematics - Abstract
In this paper, we introduce a set-valued variational inclusion problem with fuzzy mappings involving XOR-operation. We define a resolvent operator involving a bi-mapping and prove resolvent operator is single-valued, comparison and Lipschitz-type continuous. Based on resolvent operator we proposed an iterative algorithm to find the approximate solution of our problem. An existence and convergence result is proved for set-valued variational inclusion problem with fuzzy mappings involving XOR-operation without using the properties of a normal cone. Examples are constructed for illustration.
- Published
- 2021
36. Lyapunov-type inequalities for Hadamard fractional differential equation under Sturm-Liouville boundary conditions
- Author
-
Yang Zhang, Youyu Wang, and Lu Zhang
- Subjects
Lyapunov function ,hadamard fractional derivative ,General Mathematics ,lcsh:Mathematics ,Mathematics::Classical Analysis and ODEs ,green's function ,Sturm–Liouville theory ,Type (model theory) ,Mathematics::Spectral Theory ,lcsh:QA1-939 ,symbols.namesake ,Hadamard transform ,Green's function ,boundary value problem ,symbols ,Applied mathematics ,Mathematics::Metric Geometry ,Cover (algebra) ,Boundary value problem ,Fractional differential ,lyapunov-type inequality ,Mathematics - Abstract
In this paper, we establish new Lyapunov-type inequalities for a Hadamard fractional differential equation under Sturm-Liouville boundary conditions. Our conclusions cover many results in the literature.
- Published
- 2021
37. Numerical solution of fractional differential equations with temporal two-point BVPs using reproducing kernal Hilbert space method
- Author
-
Banan Maayah, Yassamine Chellouf, Salam Alnabulsi, Shaher Momani, and Ahmad Alawneh
- Subjects
General Mathematics ,Numerical analysis ,lcsh:Mathematics ,Hilbert space ,numerical method ,reproducing kernel hilbert space method (rkhsm) ,fractional differential equations ,lcsh:QA1-939 ,Fractional calculus ,symbols.namesake ,temporal two-point boundary value problems ,symbols ,Applied mathematics ,Point (geometry) ,Boundary value problem ,Fractional differential ,Approximate solution ,approximate solution ,Mathematics ,Reproducing kernel Hilbert space - Abstract
In this paper, the reproducing kernel Hilbert space method had been extended to model a numerical solution with two-point temporal boundary conditions for the fractional derivative in the Caputo sense, convergent analysis and error bounds are discussed to verify the theoretical results. Numerical examples are given to illustrate the accuracy and efficiency of the presented approach.
- Published
- 2021
38. Mild solutions for a multi-term fractional differential equation via resolvent operators
- Author
-
Rodrigo Ponce and Yong-Kui Chang
- Subjects
nonlocal problem ,fractional diffusion equation ,caputo fractional derivatives ,General Mathematics ,lcsh:Mathematics ,Applied mathematics ,Fractional differential ,lcsh:QA1-939 ,Mathematics ,Term (time) ,Resolvent ,resolvent operators - Abstract
This paper is concerned with multi-term fractional differential equations. With the help of the theory of fractional resolvent families, we establish the existence of mild solutions to a multi-term fractional differential equation.
- Published
- 2021
39. Relaxed stability conditions for linear systems with time-varying delays via some novel approaches
- Author
-
Myeong-Jin Park, Seung-Hoon Lee, Oh-Min Kwon, Young-Jae Kim, and Yong-gwon Lee
- Subjects
Lemma (mathematics) ,Stability criterion ,General Mathematics ,lcsh:Mathematics ,Linear system ,Zero (complex analysis) ,Auxiliary function ,stability ,lcsh:QA1-939 ,Stability (probability) ,lmis ,time-varying delay ,Relaxed stability ,Applied mathematics ,linear system ,lyapunov sense ,Mathematics - Abstract
In this paper, the stability problem with considering time-varying delays of linear systems is investigated. By constructing new augmented Lyapunov-Krasovskii (L-K) functionals based on auxiliary function-based integral inequality (AFBI) and considering Finsler's lemma, a stability criterion is derived. Based on the previous result, a less conservative result is proposed through the augmented zero equality approach. Finally, numerical examples are given to show the effect of the proposed criteria.
- Published
- 2021
40. The extension of analytic solutions to FDEs to the negative half-line
- Author
-
Minvydas Ragulskis, Romas Marcinkevicius, Zenonas Navickas, Inga Timofejeva, Tadas Telksnys, and Xiao-Jun Yang
- Subjects
Series (mathematics) ,inverse balancing ,General Mathematics ,lcsh:Mathematics ,Characteristic equation ,solitary wave ,negative half-line ,Extension (predicate logic) ,lcsh:QA1-939 ,Fractional power ,operator calculus ,Scheme (mathematics) ,fractional differential equation ,Applied mathematics ,Half line ,Fractional differential ,Operator calculus ,Mathematics - Abstract
An analytical framework for the extension of solutions to fractional differential equations (FDEs) to the negative half-line is presented in this paper. The proposed technique is based on the construction of a special characteristic equation corresponding to the original FDE (when the characteristic equation does exist). This characteristic equation enables the construction analytic solutions to FDEs are expressed in the form of infinite fractional power series. Necessary and sufficient conditions for the existence of such an extension are discussed in detail. It is demonstrated that the extension of solutions to FDEs to the negative half-line is not a single-valued operation. Computational experiments are used to illustrate the efficacy of the proposed scheme.
- Published
- 2021
41. The recurrence formula for the number of solutions of a equation in finite field
- Author
-
Yanbo Song
- Subjects
characters ,General Mathematics ,Recurrence formula ,lcsh:Mathematics ,lcsh:QA1-939 ,symbols.namesake ,guass sum ,Finite field ,analytic methods ,Gauss sum ,recurrence formula ,symbols ,Applied mathematics ,finite field ,Mathematics - Abstract
The main purpose of this paper is using analytic methods to give a recurrence formula of the number of solutions of an equation over finite field. We use analytic methods to give a recurrence formula for the number of solutions of the above equation. And our method is based on the properties of the Gauss sum. It is worth noting that we used a novel method to simplify the steps and avoid complicated calculations.
- Published
- 2021
42. Dynamic behaviors for reaction-diffusion neural networks with mixed delays
- Author
-
Mei Xu and Bo Du
- Subjects
Class (set theory) ,Property (philosophy) ,Artificial neural network ,Semigroup ,General Mathematics ,lcsh:Mathematics ,reaction-diffusion ,Dissipation ,neural networks ,lcsh:QA1-939 ,mixed delays ,Exponential function ,Exponential stability ,Reaction–diffusion system ,Applied mathematics ,Mathematics - Abstract
A class of reaction-diffusion neural networks with mixed delays is studied. We will discuss some important properties of the periodic mild solutions including existence and globally exponential stability by using exponential dissipation property of semigroup of operators and some analysis techniques. Finally, an example for the above neural networks is given to show the effectiveness of the results in this paper.
- Published
- 2020
43. Relation-theoretic fixed point theorems under a new implicit function with applications to ordinary differential equations
- Author
-
Abdullah Aldurayhim, Mohammad Imdad, Waleed M. Alfaqih, and Atiya Perveen
- Subjects
binary relation ,implicit function ,Implicit function ,Relation (database) ,Binary relation ,General Mathematics ,lcsh:Mathematics ,Fixed-point theorem ,multidimensional fixed point ,Fixed point ,lcsh:QA1-939 ,fixed point ,Ordinary differential equation ,ordinary differential equations ,Applied mathematics ,Uniqueness ,Mathematics - Abstract
In this paper, we introduce a new implicit function without any continuity requirement and utilize the same to prove unified relation-theoretic fixed point results. We adopt some examples to exhibit the utility of our implicit function. Furthermore, we use our results to derive some multidimensional fixed point results. Finally, as applications of our results, we study the existence and uniqueness of solution for a first-order ordinary differential equations.
- Published
- 2020
44. Integral transforms of an extended generalized multi-index Bessel function
- Author
-
Thabet Abdeljawad, Kottakkaran Sooppy Nisar, Rana Safdar Ali, Iqra Nayab, Shahid Mubeen, and Gauhar Rahman
- Subjects
Laplace transform ,extended beta transform ,General Mathematics ,Operator (physics) ,lcsh:Mathematics ,Mathematics::Classical Analysis and ODEs ,Function (mathematics) ,Extension (predicate logic) ,Integral transform ,lcsh:QA1-939 ,symbols.namesake ,appell function ,Kernel (statistics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,symbols ,Laguerre polynomials ,Applied mathematics ,extended multi-index bessel function ,fractional integrals and derivatives ,Bessel function ,Mathematics - Abstract
In this paper, we discuss the extended generalized multi-index Bessel function by using the extended beta type function. Then we investigate its several properties including integral representation, derivatives, beta transform, Laplace transform, Mellin transforms, and some relations of extension of extended generalized multi-index Bessel function (E1GMBF) with the Laguerre polynomial and Whittaker functions. Further, we also discuss the composition of the generalized fractional integral operator having Appell function as a kernel with the extension of extended generalized multi-index Bessel function and establish these results in terms of Wright functions.
- Published
- 2020
45. Novel stability criteria on a patch structure Nicholson’s blowflies model with multiple pairs of time-varying delays
- Author
-
Xin Long
- Subjects
Equilibrium point ,nicholson’s blowflies model ,convergence ,Exponential convergence ,General Mathematics ,lcsh:Mathematics ,Zero (complex analysis) ,Structure (category theory) ,asymptotical stability ,Dynamical system ,Mathematical proof ,lcsh:QA1-939 ,Stability (probability) ,patch structure ,time-varying delay ,Convergence (routing) ,Applied mathematics ,Mathematics - Abstract
This paper investigates a patch structure Nicholson's blowflies model involving multiple pairs of different time-varying delays. Without assuming the uniform positiveness of the death rate and the boundedness of coefficients, we establish three novel criteria to check the global convergence, generalized exponential convergence and asymptotical stability on the zero equilibrium point of the addressed model, respectively. Our proofs make substantial use of differential inequality techniques and dynamical system approaches, and the obtained results improve and supplement some existing ones. Last but not least, a numerical example with its simulations is given to show the feasibility of the theoretical results.
- Published
- 2020
46. Existence of infinitely many solutions for a nonlocal problem
- Author
-
Jing Yang
- Subjects
Class (set theory) ,Reduction (recursion theory) ,Property (philosophy) ,critical exponent ,General Mathematics ,lcsh:Mathematics ,fractional laplacian ,lcsh:QA1-939 ,Term (time) ,Arbitrarily large ,Applied mathematics ,reduction method ,Fractional Laplacian ,Henon equation ,Critical exponent ,Mathematics - Abstract
In this paper, we deal with a class of fractional Henon equation and by using the Lyapunov-Schmidt reduction method, under some suitable assumptions, we derive the existence of infinitely many solutions, whose energy can be made arbitrarily large. Compared to the previous works, we encounter some new challenges because of the nonlocal property for fractional Laplacian. But by doing some delicate estimates for the nonlocal term we overcome the difficulty and find infinitely many nonradial solutions.
- Published
- 2020
47. Multistep hybrid viscosity method for split monotone variational inclusion and fixed point problems in Hilbert spaces
- Author
-
Poom Kumam, Jamilu Abubakar, and Jitsupa Deepho
- Subjects
Generalization ,Iterative method ,fixed point problem ,General Mathematics ,lcsh:Mathematics ,split monotone variational inclusion ,Solution set ,Hilbert space ,variational inequality problem ,Fixed point ,lcsh:QA1-939 ,symbols.namesake ,Monotone polygon ,Variational inequality ,triple hierarchical variational inequality problem ,symbols ,Applied mathematics ,Hierarchical control system ,hilbert spaces ,Mathematics - Abstract
In this paper, we present a multi-step hybrid iterative method. It is proven that under appropriate assumptions, the proposed iterative method converges strongly to a common element of fixed point of a finite family of nonexpansive mappings, the solution set of split monotone variational inclusion problem and the solution set of triple hierarchical variational inequality problem (THVI) in real Hilbert spaces. In addition, we give a numerical example of a triple hierarchical system derived from our generalization.
- Published
- 2020
48. Lipschitz stability of an inverse problem for the Kawahara equation with damping
- Author
-
Arivazhagan Anbu, Sakthivel Kumarasamy, and Barani Balan Natesan
- Subjects
inverse problems ,General Mathematics ,lcsh:Mathematics ,carleman estimate ,Mathematics::Analysis of PDEs ,Stability result ,Type (model theory) ,Inverse problem ,Mathematics::Spectral Theory ,stability ,Lipschitz continuity ,lcsh:QA1-939 ,Stability (probability) ,kawahara equation ,Applied mathematics ,Boundary value problem ,Uniqueness ,Mathematics - Abstract
The aim of this paper is to establish a stability result regarding the inverse problem of retrieving the damping coefficient in Kawahara equation. We first establish an internal Carleman estimate for the linearized problem with the help of Dirichlet-Neumann type boundary conditions. Using the obtained Carleman estimate and the regularity of solutions for the Kawahara equation, we prove the Lipschitz type stability and uniqueness of the considered inverse problems.
- Published
- 2020
49. Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type
- Author
-
Mohammed S. Abdo, Saleh S. Redhwan, and Sadikali L. Shaikh
- Subjects
fractional gronwall inequality ,Mathematics::Functional Analysis ,ulam-hyers stability ,General Mathematics ,katugampola fractional operator ,lcsh:Mathematics ,Mathematics::Classical Analysis and ODEs ,Fixed-point theorem ,fractional differential equations ,Type (model theory) ,lcsh:QA1-939 ,fixed point theorems ,Nonlinear system ,Gronwall's inequality ,Applied mathematics ,Periodic boundary conditions ,Boundary value problem ,Uniqueness ,Fractional differential ,Mathematics - Abstract
This paper deals with a nonlinear implicit fractional differential equation with the anti-periodic boundary condition involving the Caputo-Katugampola type. The existence and uniqueness results are established by applying the fixed point theorems of Krasnoselskii and Banach. Further, by using generalized Gronwall inequality the Ulam-Hyers stability results are proved. To demonstrate the effectiveness of the main results, appropriate examples are granted.
- Published
- 2020
50. Random attractors of the stochastic extended Brusselator system with a multiplicative noise
- Author
-
Jie Xin, Chunting Ji, and Hui Liu
- Subjects
multiplicative noise ,random attractors ,pullback asymptotic compactness ,General Mathematics ,lcsh:Mathematics ,Coupling (probability) ,lcsh:QA1-939 ,Multiplicative noise ,stochastic extended brusselator system ,Brusselator ,Compact space ,Pullback ,Attractor ,Applied mathematics ,Scaling ,Brownian motion ,Mathematics - Abstract
In this paper, we are devoted to study asymptotic dynamics of the stochastic extended Brusselator system with a multiplicative noise. The stochastic extended Brusselator system is composed of three pairs of symmetrical coupling components. We firstly study the pullback absorbing property for the stochastic extended Brusselator system with a multiplicative noise. But coupling terms bring great difficulty on this problem, we use the scaling method and estimate groups to overcome this difficulty. Then, we apply the bootstrap pullback estimations to prove the pullback asymptotic compactness for the stochastic extended Brusselator system with a multiplicative noise. Finally, we show the existence of random attractors. In the study of the existence of random attractors for stochastic dynamics, we use the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms. Based on this, we have to estimate groups to overcome the difficulties of coupling structure and make more complex estimates.
- Published
- 2020
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