117 results
Search Results
2. Error estimates of variational discretization for semilinear parabolic optimal control problems
- Author
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Zuliang Lu, Xuejiao Chen, Chunjuan Hou, and Fei Huang
- Subjects
Discretization ,General Mathematics ,lcsh:Mathematics ,Type (model theory) ,semilinear parabolic equations ,Residual ,Optimal control ,lcsh:QA1-939 ,Backward Euler method ,Omega ,Finite element method ,error estimates ,optimal control problems ,A priori and a posteriori ,Applied mathematics ,finite element methods ,Mathematics - Abstract
In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)$ using backward Euler method for standard finite element. In this paper, the better result $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ is gained. Beyond that, we get a posteriori error estimates of residual type.
- Published
- 2021
3. Two new preconditioners for mean curvature-based image deblurring problem
- Author
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Rashad Ahmed, Adel M. Al-Mahdi, and Shahbaz Ahmad
- Subjects
Deblurring ,Discretization ,numerical analysis ,Computer science ,General Mathematics ,Numerical analysis ,mean curvature ,Krylov subspace ,ill-posed problem ,image deblurring ,Nonlinear system ,Fixed-point iteration ,preconditioning ,Computer Science::Computer Vision and Pattern Recognition ,Convergence (routing) ,QA1-939 ,Applied mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The mean curvature-based image deblurring model is widely used to enhance the quality of the deblurred images. However, the discretization of the associated Euler-Lagrange equations produce a nonlinear ill-conditioned system which affect the convergence of the numerical algorithms like Krylov subspace methods. To overcome this difficulty, in this paper, we present two new symmetric positive definite (SPD) preconditioners. An efficient algorithm is presented for the mean curvature-based image deblurring problem which combines a fixed point iteration (FPI) with new preconditioned matrices to handle the nonlinearity and ill-conditioned nature of the large system. The eigenvalues analysis is also presented in the paper. Fast convergence has shown in the numerical results by using the proposed new preconditioners.
- Published
- 2021
4. A posteriori error estimates of spectral method for the fractional optimal control problems with non-homogeneous initial conditions
- Author
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Xingyang Ye and Chuanju Xu
- Subjects
Spacetime ,Discretization ,General Mathematics ,a posteriori error ,fractional optimal control problem ,spectral method ,State (functional analysis) ,initial conditions ,Optimal control ,Non homogeneous ,Fractional diffusion ,QA1-939 ,A priori and a posteriori ,Applied mathematics ,Spectral method ,Mathematics - Abstract
In this paper we consider an optimal control problem governed by a space-time fractional diffusion equation with non-homogeneous initial conditions. A spectral method is proposed to discretize the problem in both time and space directions. The contribution of the paper is threefold: (1) A discussion and better understanding of the initial conditions for fractional differential equations with Riemann-Liouville and Caputo derivatives are presented. (2) A posteriori error estimates are obtained for both the state and the control approximations. (3) Numerical experiments are performed to verify that the obtained a posteriori error estimates are reliable.
- Published
- 2021
5. Airplane designing using Quadratic Trigonometric B-spline with shape parameters
- Author
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Abdul Majeed, Yushalify Misro, Mohsin Kamran, Amna Abdul Sittar, and Muhammad Abbas
- Subjects
Airfoil ,Computer science ,General Mathematics ,B-spline ,uniform knots ,Field (mathematics) ,Vertical stabilizer ,airplane parts ,open curves ,Spline (mathematics) ,Quadratic equation ,Computer Science::Graphics ,shape parameters ,QA1-939 ,Applied mathematics ,quadratic trigonometric b-spline functions ,closed curves ,Trigonometry ,Mathematics ,Free parameter ,curve designing - Abstract
The B-spline curves have been grasped tremendous achievements inside the widely identified field of Computer Aided Geometric Design (CAGD). In CAGD, spline functions have been used for the designing of various objects. In this paper, new Quadratic Trigonometric B-spline (QTBS) functions with two shape parameters are introduced. The proposed QTBS functions inherit the basic properties of classical B-spline and have been proved in this paper. The proposed scheme associated with two shape parameters where the classical B-spline functions do not have. The QTBS has been used for designing of different parts of airplane like winglet, airfoil, turbo-machinery blades and vertical stabilizer. The designed part can be controlled or changed using free parameters. The effect of shape parameters is also expressed.
- Published
- 2021
6. On the supporting nodes in the localized method of fundamental solutions for 2D potential problems with Dirichlet boundary condition
- Author
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Zengtao Chen and Fajie Wang
- Subjects
Computer science ,General Mathematics ,Selection strategy ,Stability (learning theory) ,localized method of fundamental solutions ,symbols.namesake ,Simple (abstract algebra) ,Dirichlet boundary condition ,Empirical formula ,Curve fitting ,symbols ,empirical formula ,QA1-939 ,Applied mathematics ,Method of fundamental solutions ,Node (circuits) ,meshless method ,supporting nodes ,potential problems ,Mathematics - Abstract
This paper proposes a simple, accurate and effective empirical formula to determine the number of supporting nodes in a newly-developed method, the localized method of fundamental solutions (LMFS). The LMFS has the merits of meshless, high-accuracy and easy-to-simulation in large-scale problems, but the number of supporting nodes has a certain impact on the accuracy and stability of the scheme. By using the curve fitting technique, this study established a simple formula between the number of supporting nodes and the node spacing. Based on the developed formula, the reasonable number of supporting nodes can be determined according to the node spacing. Numerical experiments confirmed the validity of the proposed methodology. This paper perfected the theory of the LMFS, and provided a quantitative selection strategy of method parameters.
- Published
- 2021
7. 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
8. 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
9. 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
10. 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
11. 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
12. 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
13. A new algorithm based on compressed Legendre polynomials for solving boundary value problems
- Author
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Yingzhen Lin, Hui Zhu, and Liangcai Mei
- Subjects
compressed legendre polynomials ,boundary value problems ,General Mathematics ,error estimation ,QA1-939 ,Applied mathematics ,Boundary value problem ,stability analysis ,Legendre polynomials ,Mathematics ,convergence analysis - Abstract
In this paper, we discuss a novel numerical algorithm for solving boundary value problems. We introduce an orthonormal basis generated from compressed Legendre polynomials. This basis can avoid Runge phenomenon caused by high-order polynomial approximation. Based on the new basis, a numerical algorithm of two-point boundary value problems is established. The convergence and stability of the method are proved. The whole analysis is also applicable to higher order equations or equations with more complex boundary conditions. Four numerical examples are tested to illustrate the accuracy and efficiency of the algorithm. The results show that our algorithm have higher accuracy for solving linear and nonlinear problems.
- Published
- 2022
14. Elastic transformation method for solving ordinary differential equations with variable coefficients
- Author
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Xiaoxu Dong, Shunchu Li, Pengshe Zheng, and Jing Luo
- Subjects
Transformation (function) ,General Mathematics ,Ordinary differential equation ,variable coefficient ,general solution ,elastic transformation method ,QA1-939 ,Applied mathematics ,laguerre equation ,ordinary differential equation ,Mathematics ,Variable (mathematics) - Abstract
Aiming at the problem of solving nonlinear ordinary differential equations with variable coefficients, this paper introduces the elastic transformation method into the process of solving ordinary differential equations for the first time. A class of first-order and a class of third-order ordinary differential equations with variable coefficients can be transformed into the Laguerre equation through elastic transformation. With the help of the general solution of the Laguerre equation, the general solution of these two classes of ordinary differential equations can be obtained, and then the curves of the general solution can be drawn. This method not only expands the solvable classes of ordinary differential equations, but also provides a new idea for solving ordinary differential equations with variable coefficients.
- Published
- 2022
15. On ψ-Hilfer generalized proportional fractional operators
- Author
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Subhash Alha, Ali Akgül, Idris Ahmed, Fahd Jarad, and Ishfaq Ahmad Mallah
- Subjects
General Mathematics ,weighed space ,QA1-939 ,Applied mathematics ,generalized proportional fractional derivative ,hilfer fractional derivative ,fixed point theorems ,Mathematics ,existence and uniqueness - Abstract
In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.
- Published
- 2022
16. New Lyapunov-type inequalities for fractional multi-point boundary value problems involving Hilfer-Katugampola fractional derivative
- Author
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Wei Zhang, Jifeng Zhang, and Jinbo Ni
- Subjects
Lyapunov function ,General Mathematics ,hilfer-katugampola fractional derivative ,Type (model theory) ,Fractional calculus ,symbols.namesake ,multi-point boundary condition ,symbols ,QA1-939 ,Applied mathematics ,Boundary value problem ,lyapunov-type inequality ,Multi point ,Mathematics - Abstract
In this paper, we present new Lyapunov-type inequalities for Hilfer-Katugampola fractional differential equations. We first give some unique properties of the Hilfer-Katugampola fractional derivative, and then by using these new properties we convert the multi-point boundary value problems of Hilfer-Katugampola fractional differential equations into the equivalent integral equations with corresponding Green's functions, respectively. Finally, we make use of the Banach's contraction principle to derive the desired results, and give a series of corollaries to show that the current results extend and enrich the previous results in the literature.
- Published
- 2022
17. On a boundary value problem for fractional Hahn integro-difference equations with four-point fractional integral boundary conditions
- Author
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Sotiris K. Ntouyas, Thanin Sitthiwirattham, and Varaporn Wattanakejorn
- Subjects
Mathematics::Functional Analysis ,boundary value problems ,General Mathematics ,Mathematics::Classical Analysis and ODEs ,existence ,Fixed-point theorem ,fractional hahn difference ,Fixed point ,Quantum number ,Nonlinear system ,Operator (computer programming) ,QA1-939 ,Applied mathematics ,fractional hahn integral ,Point (geometry) ,Boundary value problem ,Uniqueness ,Mathematics - Abstract
In this paper, we study a boundary value problem consisting of Hahn integro-difference equation supplemented with four-point fractional Hahn integral boundary conditions. The novelty of this problem lies in the fact that it contains two fractional Hahn difference operators and three fractional Hahn integrals with different quantum numbers and orders. Firstly, we convert the given nonlinear problem into a fixed point problem, by considering a linear variant of the problem at hand. Once the fixed point operator is available, we make use the classical Banach's and Schauder's fixed point theorems to establish existence and uniqueness results. An example is also constructed to illustrate the main results. Several properties of fractional Hahn integral that will be used in our study are also discussed.
- Published
- 2022
18. Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses
- Author
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Kaihong Zhao and Shuang Ma
- Subjects
Class (set theory) ,Mathematics::Functional Analysis ,General Mathematics ,Stability (learning theory) ,stability ,hadamard fractional integral bvp ,contraction mapping principle ,Nonlinear system ,Hadamard transform ,QA1-939 ,Applied mathematics ,Boundary value problem ,Mathematics ,existence and uniqueness - Abstract
This paper considers a class of nonlinear implicit Hadamard fractional differential equations with impulses. By using Banach's contraction mapping principle, we establish some sufficient criteria to ensure the existence and uniqueness of solution. Furthermore, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of this system are obtained by applying nonlinear functional analysis technique. As applications, an interesting example is provided to illustrate the effectiveness of main results.
- Published
- 2022
19. The extended Weibull–Fréchet distribution: properties, inference, and applications in medicine and engineering
- Author
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Ekramy A. Hussein, Ahmed Z. Afify, and Hassan M. Aljohani
- Subjects
Distribution (number theory) ,engineering data ,General Mathematics ,Inference ,Failure rate ,Probability density function ,maximum product of spacing estimators ,cramér–von mises estimation ,Frequentist inference ,fréchet distribution ,Generalized extreme value distribution ,QA1-939 ,Fréchet distribution ,Applied mathematics ,simulations ,Mathematics ,Weibull distribution ,extreme value distribution - Abstract
In this paper, a flexible version of the Fréchet distribution called the extended Weibull–Fréchet (EWFr) distribution is proposed. Its failure rate has a decreasing shape, an increasing shape, and an upside-down bathtub shape. Its density function can be a symmetric shape, an asymmetric shape, a reversed-J shape and J shape. Some mathematical properties of the EWFr distribution are explored. The EWFr parameters are estimated using several frequentist estimation approaches. The performance of these methods is addressed using detailed simulations. Furthermore, the best approach for estimating the EWFr parameters is determined based on partial and overall ranks. Finally, the performance of the EWFr distribution is studied using two real-life datasets from the medicine and engineering sciences. The EWFr distribution provides a superior fit over other competing Fréchet distributions such as the exponentiated-Fréchet, beta-Fréchet, Lomax–Fréchet, and Kumaraswamy Marshall–Olkin Fréchet.
- Published
- 2022
20. An efficient modified hybrid explicit group iterative method for the time-fractional diffusion equation in two space dimensions
- Author
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Nur Nadiah Abd Hamid, Fouad Mohammad Salama, Norhashidah Hj. Mohd. Ali, and Umair Ali
- Subjects
fractional diffusion equation ,Group (mathematics) ,Iterative method ,General Mathematics ,caputo fractional derivative ,Space (mathematics) ,Fractional diffusion ,QA1-939 ,Applied mathematics ,laplace transform ,stability and convergence ,grouping strategy ,finite difference scheme ,Mathematics - Abstract
In this paper, a new modified hybrid explicit group (MHEG) iterative method is presented for the efficient and accurate numerical solution of a time-fractional diffusion equation in two space dimensions. The time fractional derivative is defined in the Caputo sense. In the proposed method, a Laplace transformation is used in the temporal domain, and, for the spatial discretization, a new finite difference scheme based on grouping strategy is considered. The unique solvability, unconditional stability and convergence are thoroughly proved by the matrix analysis method. Comparison of numerical results with analytical and other approximate solutions indicates the viability and efficiency of the proposed algorithm.
- Published
- 2022
21. On stochastic accelerated gradient with non-strongly convexity
- Author
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Xingxing Zha, Yongquan Zhang, Dongyin Wang, and Yiyuan Cheng
- Subjects
least-square regression ,General Mathematics ,Carry (arithmetic) ,logistic regression ,Supervised learning ,Regular polygon ,Lipschitz continuity ,Stochastic approximation ,accelerated stochastic approximation ,Convexity ,Stochastic programming ,convergence rate ,Rate of convergence ,QA1-939 ,Applied mathematics ,Mathematics - Abstract
In this paper, we consider stochastic approximation algorithms for least-square and logistic regression with no strong-convexity assumption on the convex loss functions. We develop two algorithms with varied step-size motivated by the accelerated gradient algorithm which is initiated for convex stochastic programming. We analyse the developed algorithms that achieve a rate of $ O(1/n^{2}) $ where $ n $ is the number of samples, which is tighter than the best convergence rate $ O(1/n) $ achieved so far on non-strongly-convex stochastic approximation with constant-step-size, for classic supervised learning problems. Our analysis is based on a non-asymptotic analysis of the empirical risk (in expectation) with less assumptions that existing analysis results. It does not require the finite-dimensionality assumption and the Lipschitz condition. We carry out controlled experiments on synthetic and some standard machine learning data sets. Empirical results justify our theoretical analysis and show a faster convergence rate than existing other methods.
- Published
- 2022
22. 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
23. 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
- Subjects
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
24. 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
25. 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.
- Published
- 2021
26. A relaxed generalized Newton iteration method for generalized absolute value equations
- Author
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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
27. 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
28. 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
29. 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
30. A basic study of a fractional integral operator with extended Mittag-Leffler kernel
- Author
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Iyad Suwan, Asad Ali, Thabet Abdeljawad, Kottakkaran Sooppy Nisar, Muhammad Samraiz, and Gauhar Rahman
- Subjects
fractional integral ,Mathematics::Complex Variables ,General Mathematics ,Operator (physics) ,Mathematics::Classical Analysis and ODEs ,Function (mathematics) ,Extension (predicate logic) ,Type (model theory) ,symbols.namesake ,Mathematics::Probability ,mittag-leffler function ,Kernel (statistics) ,Mittag-Leffler function ,prabhakar fractional integral ,symbols ,QA1-939 ,Applied mathematics ,Differential (mathematics) ,Mathematics - Abstract
In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.
- Published
- 2021
31. Maximal and minimal iterative positive solutions for p-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term
- Author
-
Lishan Liu, Ying Wang, and Limin Guo
- Subjects
General Mathematics ,Function (mathematics) ,Derivative ,Term (time) ,hadamard fractional differential equation ,Nonlinear system ,infinite-point ,Hadamard transform ,positive solution ,p-Laplacian ,QA1-939 ,Applied mathematics ,Boundary value problem ,Fractional differential ,Mathematics ,iterative positive solution - Abstract
In this paper, the maximal and minimal iterative positive solutions are investigated for a singular Hadamard fractional differential equation boundary value problem with a boundary condition involving values at infinite number of points. Green's function is deduced and some properties of Green's function are given. Based upon these properties, iterative schemes are established for approximating the maximal and minimal positive solutions.
- Published
- 2021
32. Modelling chaotic dynamical attractor with fractal-fractional differential operators
- Author
-
Youssef El-Khatib and Sonal Jain
- Subjects
Computer science ,General Mathematics ,Chaotic ,Differential operator ,Dynamical system ,Convolution ,Mathematical Operators ,fractal-fractional integral operator ,Fractal ,Attractor ,chaotic attractors ,QA1-939 ,Applied mathematics ,as strange attractor ,Differential (mathematics) ,fractal-fractional differential operators ,Mathematics - Abstract
Differential operators based on convolution have been recognized as powerful mathematical operators able to depict and capture chaotic behaviors, especially those that are not able to be depicted using classical differential and integral operators. While these differential operators have being applied with great success in many fields of science, especially in the case of dynamical system, we have to confess that they were not able depict some chaotic behaviors, especially those with additionally similar patterns. To solve this issue new class of differential and integral operators were proposed and applied in few problems. In this paper, we aim to depict chaotic behavior using the newly defined differential and integral operators with fractional order and fractal dimension. Additionally we introduced a new chaotic operators with strange attractors. Several simulations have been conducted and illustrations of the results are provided to show the efficiency of the models.
- Published
- 2021
33. A high order numerical method for solving Caputo nonlinear fractional ordinary differential equations
- Author
-
Xumei Zhang and Junying Cao
- Subjects
higher order numerical scheme ,General Mathematics ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,caputo derivative ,nonlinear fractional ordinary differential equations ,convergence analysis ,Nonlinear system ,Ordinary differential equation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,QA1-939 ,Order (group theory) ,Applied mathematics ,finite difference method ,Mathematics - Abstract
In this paper, we construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations. Firstly, we use the piecewise Quadratic Lagrange interpolation method to construct a high order numerical scheme for Caputo nonlinear fractional ordinary differential equations, and then analyze the local truncation error of the high order numerical scheme. Secondly, based on the local truncation error, the convergence order of $ 3-\theta $ order is obtained. And the convergence are strictly analyzed. Finally, the numerical simulation of the high order numerical scheme is carried out. Through the calculation of typical problems, the effectiveness of the numerical algorithm and the correctness of theoretical analysis are verified.
- Published
- 2021
34. Analysis of 2D heat conduction in nonlinear functionally graded materials using a local semi-analytical meshless method
- Author
-
Yanpeng Gong, Fajie Wang, and Chao Wang
- Subjects
Helmholtz equation ,General Mathematics ,local knot method ,Inverse ,Basis function ,heat conduction ,Boundary knot method ,System of linear equations ,Thermal conduction ,Functionally graded material ,Nonlinear system ,QA1-939 ,Applied mathematics ,semi-analytical meshless method ,nonlinear functionally graded material ,Mathematics ,kirchhoff transformation - Abstract
This paper proposes a local semi-analytical meshless method for simulating heat conduction in nonlinear functionally graded materials. The governing equation of heat conduction problem in nonlinear functionally graded material is first transformed to an anisotropic modified Helmholtz equation by using the Kirchhoff transformation. Then, the local knot method (LKM) is employed to approximate the solution of the transformed equation. After that, the solution of the original nonlinear equation can be obtained by the inverse Kirchhoff transformation. The LKM is a recently proposed meshless approach. As a local semi-analytical meshless approach, it uses the non-singular general solution as the basis function and has the merits of simplicity, high accuracy, and easy-to-program. Compared with the traditional boundary knot method, the present scheme avoids an ill-conditioned system of equations, and is more suitable for large-scale simulations associated with complicated structures. Three benchmark numerical examples are provided to confirm the accuracy and validity of the proposed approach.
- Published
- 2021
35. Oscillation theorems of solution of second-order neutral differential equations
- Author
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Omar Bazighifan, Hammad Alotaibi, Ali Muhib, and Kamsing Nonlaopon
- Subjects
Class (set theory) ,Oscillation ,Differential equation ,General Mathematics ,second-order neutral differential equation ,QA1-939 ,Applied mathematics ,Order (group theory) ,neutral differential equation ,oscillation criteria ,Neutral differential equations ,Mathematics - Abstract
In this paper, we aim to explore the oscillation of solutions for a class of second-order neutral functional differential equations. We propose new criteria to ensure that all obtained solutions are oscillatory. The obtained results can be used to develop and provide theoretical support for and further develop the oscillation study for a class of second-order neutral differential equations. Finally, an illustrated example is given to demonstrate the effectiveness of our new criteria.
- Published
- 2021
36. On solvability of some p-Laplacian boundary value problems with Caputo fractional derivative
- Author
-
Dexin Chen and Xiaoping Li
- Subjects
General Mathematics ,boundary value problem ,caputo fractional derivative ,p-Laplacian ,QA1-939 ,Fixed-point theorem ,Applied mathematics ,fixed point theorem ,Boundary value problem ,solvability ,Mathematics ,Fractional calculus - Abstract
The solvability of some $ p $-Laplace boundary value problems with Caputo fractional derivative are discussed. By using the fixed-point theory and analysis techniques, some existence results of one or three non-negative solutions are obtained. Two examples showed that the conditions used in this paper are somewhat easy to check.
- Published
- 2021
37. Fixed point results of an implicit iterative scheme for fractal generations
- Author
-
Muhammad Tanveer, Yi-Xia Li, Qingxiu Peng, Haixia Zhang, and Nehad Ali Shah
- Subjects
jungck-ishikawa iteration ,Fractal ,General Mathematics ,Scheme (mathematics) ,fixed point theory ,fractals ,QA1-939 ,Applied mathematics ,Fixed point ,Mathematics - Abstract
In this paper, we derive the escape criteria for general complex polynomial $ f(x) = \sum_{i = 0}^{p}a_{i}x^{i} $ with $ p\geq2 $, where $ a_{i} \in \mathbb{C} $ for $ i = 0, 1, 2, \dots, p $ to generate the fractals. Moreover, we study the orbit of an implicit iteration (i.e., Jungck-Ishikawa iteration with $ s $-convexity) and develop algorithms for Mandelbrot set and Multi-corn or Multi-edge set. Moreover, we draw some complex graphs and observe how the graph of Mandelbrot set and Multi-corn or Multi-edge set vary with the variation of $ a_{i} $'s.
- Published
- 2021
38. A class of explicit implicit alternating difference schemes for generalized time fractional Fisher equation
- Author
-
Xiao Qin, Xiaozhong Yang, and Peng Lyu
- Subjects
Class (set theory) ,convergence ,General Mathematics ,Order (ring theory) ,Fisher equation ,explicit implicit alternating difference scheme ,stability ,Stability (probability) ,Alpha (programming language) ,Singularity ,Scheme (mathematics) ,generalized time fractional fisher equation ,Convergence (routing) ,QA1-939 ,Applied mathematics ,numerical experiments ,Mathematics - Abstract
The generalized time fractional Fisher equation is one of the significant models to describe the dynamics of the system. The study of effective numerical techniques for the equation has important scientific significance and application value. Based on the alternating technique, this article combines the classical explicit difference scheme and the implicit difference scheme to construct a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation. The unconditional stability and convergence with order $ O\left({\tau }^{2-\alpha }+{h}^{2}\right) $ of the proposed schemes are analyzed. Numerical examples are performed to verify the theoretical analysis. Compared with the classical implicit difference scheme, the calculation cost of the explicit implicit alternating difference schemes is reduced by almost $ 60 $%. Numerical experiments show that the explicit implicit alternating difference schemes are also suitable for solving the time fractional Fisher equation with initial weak singularity and have an accuracy of order $ O\left({\tau }^{\alpha }+{h}^{2}\right) $, which verify that the methods proposed in this paper are efficient for solving the generalized time fractional Fisher equation.
- Published
- 2021
39. General fixed-point method for solving the linear complementarity problem
- Author
-
Xi-Ming Fang
- Subjects
algorithm ,convergence ,Iterative method ,General Mathematics ,Numerical analysis ,Diagonal ,Positive-definite matrix ,linear complementarity problem ,Linear complementarity problem ,Matrix (mathematics) ,iterative method ,Fixed-point iteration ,QA1-939 ,Applied mathematics ,solution ,Equivalence (measure theory) ,Mathematics - Abstract
In this paper, we consider numerical methods for the linear complementarity problem (LCP). By introducing a positive diagonal parameter matrix, the LCP is transformed into an equivalent fixed-point equation and the equivalence is proved. Based on such equation, the general fixed-point (GFP) method with two cases are proposed and analyzed when the system matrix is a $ P $-matrix. In addition, we provide several concrete sufficient conditions for the proposed method when the system matrix is a symmetric positive definite matrix or an $ H_{+} $-matrix. Meanwhile, we discuss the optimal case for the proposed method. The numerical experiments show that the GFP method is effective and practical.
- Published
- 2021
40. Theoretical and numerical stability results for a viscoelastic swelling porous-elastic system with past history
- Author
-
Mohammad M. Al-Gharabli, Mohamed Alahyane, and Adel M. Al-Mahdi
- Subjects
convex functions ,General Mathematics ,Relaxation (iterative method) ,swelling porous problem ,Viscoelasticity ,Term (time) ,viscoelastic ,Kernel (statistics) ,medicine ,QA1-939 ,Applied mathematics ,finite element and crank-nicolson methods ,Swelling ,medicine.symptom ,general decay ,Convex function ,Porous medium ,Mathematics ,Numerical stability - Abstract
The purpose of this paper is to establish a general stability result for a one-dimensional linear swelling porous-elastic system with past history, irrespective of the wave speeds of the system. First, we establish an explicit and general decay result under a wider class of the relaxation (kernel) functions. The kernel in our memory term is more general and of a broader class. Further, we get a better decay rate without imposing some assumptions on the boundedness of the history data considered in many earlier results in the literature. We also perform several numerical tests to illustrate our theoretical results. Our output extends and improves some of the available results on swelling porous media in the literature.
- Published
- 2021
41. Two-person zero-sum stochastic games with varying discount factors
- Author
-
Yinying Kong, Xiao Wu, Qi Wang, and Economics, Guangzhou, China
- Subjects
Computer Science::Computer Science and Game Theory ,Markov chain ,Banach fixed-point theorem ,General Mathematics ,Zero (complex analysis) ,varying discount factors ,Function (mathematics) ,Space (mathematics) ,Action (physics) ,two-person zero-sum stochastic games ,QA1-939 ,Applied mathematics ,State space ,expected discount criterion ,Optimal criterion ,Mathematics - Abstract
In this paper, two-person zero-sum Markov games with Borel state space and action space, unbounded reward function and state-dependent discount factors are studied. The optimal criterion is expected discount criterion. Firstly, sufficient conditions for the existence of optimal policies are given for the two-person zero-sum Markov games with varying discount factors. Then, the existence of optimal policies is proved by Banach fixed point theorem. Finally, we give an example for reservoir operations to illustrate the existence results.
- Published
- 2021
42. Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food
- Author
-
Hang Zheng and Luoyi Wu
- Subjects
Hopf bifurcation ,Correctness ,General Mathematics ,Functional response ,periodic solution ,global hopf bifurcation ,stability ,Critical value ,Stability (probability) ,symbols.namesake ,Normal form theory ,symbols ,QA1-939 ,Applied mathematics ,delayed predator-prey system ,local hopf bifurcation ,Center manifold ,Mathematics - Abstract
In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.
- Published
- 2021
43. Merit functions for absolute value variational inequalities
- Author
-
Khalida Inayat Noor, Muhammad Aslam Noor, and Safeera Batool
- Subjects
Class (set theory) ,absolute value variational inequalities ,General Mathematics ,fixed points ,Absolute value (algebra) ,error bounds ,Fixed point ,Complementarity theory ,Variational inequality ,Absolute value equation ,merit functions ,QA1-939 ,Applied mathematics ,Mathematics - Abstract
This article deals with a class of variational inequalities known as absolute value variational inequalities. Some new merit functions for the absolute value variational inequalities are established. Using these merit functions, we derive the error bounds for absolute value variational inequalities. Since absolute value variational inequalities contain variational inequalities, absolute value complementarity problem and system of absolute value equations as special cases, the findings presented here recapture various known results in the related domains. The conclusions of this paper are more comprehensive and may provoke futuristic research.
- Published
- 2021
44. Value functions in a regime switching jump diffusion with delay market model
- Author
-
Jose Maria L. Escaner and Dennis Llemit
- Subjects
Partial differential equation ,Stochastic process ,delay ,General Mathematics ,Jump diffusion ,Hamilton–Jacobi–Bellman equation ,optimal portfolio ,regime switching ,jump diffusion ,Dynamic programming ,value function ,Bellman equation ,Isoelastic utility ,QA1-939 ,Applied mathematics ,Stochastic optimization ,Mathematics - Abstract
In this paper, we consider a market model where the risky asset is a jump diffusion whose drift, volatility and jump coefficients are influenced by market regimes and history of the asset itself. Since the trajectory of the risky asset is discontinuous, we modify the delay variable so that it remains defined in this discontinuous setting. Instead of the actual path history of the risky asset, we consider the continuous approximation of its trajectory. With this modification, the delay variable, which is a sliding average of past values of the risky asset, no longer breaks down. We then use the resulting stochastic process in formulating the state variable of a portfolio optimization problem. In this formulation, we obtain the dynamic programming principle and Hamilton Jacobi Bellman equation. We also provide a verification theorem to guarantee the optimal solution of the corresponding stochastic optimization problem. We solve the resulting finite time horizon control problem and show that close form solutions of the stochastic optimization problem exist for the cases of power and logarithmic utility functions. In particular, we show that the HJB equation for the power utility function is a first order linear partial differential equation while that of the logarithmic utility function is a linear ordinary differential equation.
- Published
- 2021
45. Rotational periodic solutions for fractional iterative systems
- Author
-
Yi Cheng, Rui Wu, and Ravi P. Agarwal
- Subjects
fractional iterative systems ,Artificial neural network ,neural network ,General Mathematics ,010102 general mathematics ,existence ,Fixed-point theorem ,Topological degree theory ,Nonlinear control ,01 natural sciences ,Fractional calculus ,Term (time) ,010101 applied mathematics ,Nonlinear system ,QA1-939 ,Applied mathematics ,Uniqueness ,0101 mathematics ,rotational periodic ,Mathematics - Abstract
In this paper, we devoted to deal with the rotational periodic problem of some fractional iterative systems in the sense of Caputo fractional derivative. Under one sided-Lipschtiz condition on nonlinear term, the existence and uniqueness of solution for a fractional iterative equation is proved by applying the Leray-Schauder fixed point theorem and topological degree theory. Furthermore, the well posedness for a nonlinear control system with iteration term and a multivalued disturbance is established by using set-valued theory. The existence of solutions for a iterative neural network system is demonstrated at the end.
- Published
- 2021
46. Application of fractional differential equation in economic growth model: A systematic review approach
- Author
-
Jumadil Saputra, Asep K. Supriatna, Muhamad Deni Johansyah, and Endang Rusyaman
- Subjects
Differential equation ,General Mathematics ,memory effect modeling ,Economic growth model ,fractional order derivative (fde) ,differential equation ,Nonlinear system ,economic growth model ,QA1-939 ,Applied mathematics ,Order (group theory) ,Development (differential geometry) ,Uniqueness ,Fractional differential ,Approximate solution ,fractional riccati differential equation (frde) ,Mathematics - Abstract
In this paper we review the applications of fractional differential equation in economic growth models. This includes the theories about linear and nonlinear fractional differential equation, including the Fractional Riccati Differential Equation (FRDE) and its applications in economic growth models with memory effect. The method used in this study is by comparing related literatures and evaluate them comprehensively. The results of this study are the chronological order of the applications of the Fractional Differential Equation (FDE) in economic growth models and the development on theories of the FDE solutions, including the FRDE forms of economic growth models. This study also provides a comparative analysis on solutions of linear and nonlinear FDE, and approximate solution of economic growth models involving memory effects using various methods. The main contribution of this research is the chonological development of the theory to find necessary and sufficient conditions to guarantee the existence and uniqueness of the FDE in economic growth and the methods to obtain the solution. Some remarks on how further researches can be done are also presented as a general conclusion.
- Published
- 2021
47. Nonlinear Fredholm integro-differential equation in two-dimensional and its numerical solutions
- Author
-
A. M. Al-Bugami
- Subjects
Work (thermodynamics) ,General Mathematics ,homotopy analysis ,010103 numerical & computational mathematics ,01 natural sciences ,fredholm integro-differential equation ,010101 applied mathematics ,Nonlinear system ,adomian decomposition ,Integro-differential equation ,Kernel (statistics) ,QA1-939 ,Applied mathematics ,0101 mathematics ,Adomian decomposition method ,Homotopy analysis method ,Mathematics - Abstract
This paper proposes a new definition of the nonlinear Fredholm integro-differential equation of the second kind with continuous kernel in two-dimensional (NT-DFIDE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions of NT-DFIDE are obtained by two powerful methods Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM). The given numerical examples showed the efficiency and accuracy of the introduced methods.
- Published
- 2021
48. A decent three term conjugate gradient method with global convergence properties for large scale unconstrained optimization problems
- Author
-
Ahmad Alhawarat, Ibtisam Masmali, and Zabidin Salleh
- Subjects
021103 operations research ,Artificial neural network ,Scale (ratio) ,Property (programming) ,General Mathematics ,0211 other engineering and technologies ,CPU time ,inexact line search ,010103 numerical & computational mathematics ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,Term (time) ,global convergence ,Conjugate gradient method ,conjugate gradient method ,Convergence (routing) ,QA1-939 ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
The conjugate gradient (CG) method is a method to solve unconstrained optimization problems. Moreover CG method can be applied in medical science, industry, neural network, and many others. In this paper a new three term CG method is proposed. The new CG formula is constructed based on DL and WYL CG formulas to be non-negative and inherits the properties of HS formula. The new modification satisfies the convergence properties and the sufficient descent property. The numerical results show that the new modification is more efficient than DL, WYL, and CG-Descent formulas. We use more than 200 functions from CUTEst library to compare the results between these methods in term of number of iterations, function evaluations, gradient evaluations, and CPU time.
- Published
- 2021
49. Blow-up of energy solutions for the semilinear generalized Tricomi equation with nonlinear memory term
- Author
-
Jianye Xia, Wenjing Zhi, and Jincheng Shi
- Subjects
General Mathematics ,Nonlinear memory ,010102 general mathematics ,Mathematics::Analysis of PDEs ,Relaxation (iterative method) ,Function (mathematics) ,Wave equation ,01 natural sciences ,Term (time) ,010101 applied mathematics ,nonlinear memory term ,semilinear hyperbolic equation ,QA1-939 ,Applied mathematics ,wave equation ,generalized tricomi operator ,0101 mathematics ,Energy (signal processing) ,blow-up ,Mathematics - Abstract
In this paper, we investigate blow-up conditions for the semilinear generalized Tricomi equation with a general nonlinear memory term in $ \mathbb{R}^n $ by using suitable functionals and employing iteration procedures. Particularly, a new combined effect from the relaxation function and the time-dependent coefficient is found.
- Published
- 2021
50. Estimating fixed points of non-expansive mappings with an application
- Author
-
Mohd Jubair, Javid Ali, Faizan Ahmad Khan, and Yeşim Saraç
- Subjects
uniformly convex banach space ,Differential equation ,General Mathematics ,fixed points ,second order ordinary differential equation ,Banach space ,Fixed point ,iterative schemes ,non-expansive mappings ,Scheme (mathematics) ,Convergence (routing) ,QA1-939 ,Applied mathematics ,Order (group theory) ,Boundary value problem ,Expansive ,Mathematics - Abstract
In this paper, we study a three step iterative scheme to estimate fixed points of non-expansive mappings in the framework of Banach spaces. Further, some convergence results are proved for such mappings. A nontrivial numerical example is presented to verify our assertions and main results. Finally, we approximate the solution of a boundary value problem of second order differential equation.
- Published
- 2021
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