Showing papers in "Abstract and Applied Analysis in 2015"
TL;DR: In this paper, the authors studied the free flow of a Casson fluid past an oscillating vertical plate with constant wall temperature and used the Laplace transform method to find the exact solutions of these equations.
Abstract: The unsteady free flow of a Casson fluid past an oscillating vertical plate with constant wall temperature has been studied. The Casson fluid model is used to distinguish the non-Newtonian fluid behaviour. The governing partial differential equations corresponding to the momentum and energy equations are transformed into linear ordinary differential equations by using nondimensional variables. Laplace transform method is used to find the exact solutions of these equations. Expressions for shear stress in terms of skin friction and the rate of heat transfer in terms of Nusselt number are also obtained. Numerical results of velocity and temperature profiles with various values of embedded flow parameters are shown graphically and their effects are discussed in detail.
35 citations
TL;DR: In this article, the existence of fixed points for multivalued mappings, under an α-ψ-contractive condition of Ciric type, in the setting of complete b-metric spaces is investigated.
Abstract: In 2012, Samet et al. introduced the notion of α-ψ-contractive mapping and gave
sufficient conditions for the existence of fixed points for this class of mappings. The purpose of our paper is
to study the existence of fixed points for multivalued mappings,
under an α-ψ-contractive condition of Ciric type, in the setting of complete b-metric spaces. An application to integral equation is given.
29 citations
TL;DR: In this article, the concept of white noise is transferred to infinite-dimensional spaces, and the theory of stochastic Sobolev type equations is developed in the spaces of differentiable noise.
Abstract: The concept of “white noise,” initially established in finite-dimensional spaces, is transferred to infinite-dimensional case. The goal of this transition is to develop the theory of stochastic Sobolev type equations and to elaborate applications of practical interest. To reach this goal the Nelson-Gliklikh derivative is introduced and the spaces of “noises” are developed. The Sobolev type equations with relatively sectorial operators are considered in the spaces of differentiable “noises.” The existence and uniqueness of classical solutions are proved. The stochastic Dzektser equation in a bounded domain with homogeneous boundary condition and the weakened Showalter-Sidorov initial condition is considered as an application.
27 citations
TL;DR: A survey of modifications based on the classic Newton's and the higher order Newton-like root finding methods for complex polynomials is presented in this article, where instead of the standard Picard's iteration several different iteration processes, described in the literature, which are called nonstandard ones, are used.
Abstract: A survey of some modifications based on the classic Newton’s and the higher order Newton-like root finding methods for complex polynomials is presented. Instead of the standard Picard’s iteration several different iteration processes, described in the literature, which we call nonstandard ones, are used. Kalantari’s visualizations of root finding process are interesting from at least three points of view: scientific, educational, and artistic. By combining different kinds of iterations, different convergence tests, and different colouring we obtain a great variety of polynomiographs. We also check experimentally that using complex parameters instead of real ones in multiparameter iterations do not destabilize the iteration process. Moreover, we obtain nice looking polynomiographs that are interesting from the artistic point of view. Real parts of the parameters alter symmetry, whereas imaginary ones cause asymmetric twisting of polynomiographs.
25 citations
TL;DR: In this article, the influence of wall slip condition on a free convection flow of an incompressible viscous fluid with heat transfer and ramped wall temperature was explored by using Laplace transform technique.
Abstract: The objective of this study is to explore the influence of wall slip condition on a free convection flow of an incompressible viscous fluid with heat transfer and ramped wall temperature. Exact solution of the problem is obtained by using Laplace transform technique. Graphical results to see the effects of Prandtl number Pr, time , and slip parameter on velocity and skin friction for the case of ramped and constant temperature of the plate are provided and discussed.
23 citations
TL;DR: In this paper, the relationship between pressure waves and different modes of closing and opening of valves is analyzed. But the results show that changes in the pressure wave profile and amplitude depend on the type of closing laws, valve closure times, and the number of polygonal segments in the closing function.
Abstract: Water hammer on transient flow of hydrogen-natural gas mixture in a horizontal pipeline is analysed to determine the relationship between pressure waves and different modes of closing and opening of valves. Four types of laws applicable to closing valve, namely, instantaneous, linear, concave, and convex laws, are considered. These closure laws describe the speed variation of the hydrogen-natural gas mixture as the valve is closing. The numerical solution is obtained using the reduced order modelling technique. The results show that changes in the pressure wave profile and amplitude depend on the type of closing laws, valve closure times, and the number of polygonal segments in the closing function. The pressure wave profile varies from square to triangular and trapezoidal shape depending on the type of closing laws, while the amplitude of pressure waves reduces as the closing time is reduced and the numbers of polygonal segments are increased. The instantaneous and convex closing laws give rise to minimum and maximum pressure, respectively.
21 citations
TL;DR: In this article, the authors show that the epidemic model can be generalized to fractional order on a consistent framework of biological behavior, and the existence and stability of equilibrium points are studied.
Abstract: This paper shows that the epidemic model, previously proposed under ordinary differential equation theory, can be generalized to fractional order on a consistent framework of biological behavior. The domain set for the model in which all variables are restricted is established. Moreover, the existence and stability of equilibrium points are studied. We present the proof that endemic equilibrium point when reproduction number is locally asymptotically stable. This result is achieved using the linearization theorem for fractional differential equations. The global asymptotic stability of disease-free point, when , is also proven by comparison theory for fractional differential equations. The numeric simulations for different scenarios are carried out and data obtained are in good agreement with theoretical results, showing important insight about the use of the fractional coupled differential equations set to model babesiosis disease and tick populations.
21 citations
TL;DR: In this article, the authors prove the controllability of the semilinear impulsive evolution equation with respect to a strongly continuous semigroup, where the linear system is controllable on for all.
Abstract: We prove the approximate controllability of the following semilinear impulsive evolution equation: where , and are Hilbert spaces, , is a bounded linear operator, are smooth functions, and is an unbounded linear operator in which generates a strongly continuous semigroup . We suppose that is bounded and the linear system is approximately controllable on for all . Under these conditions, we prove the following statement: the semilinear impulsive evolution equation is approximately controllable on .
21 citations
TL;DR: In this paper, a model predictive controller designed using a type of orthonormal functions called Laguerre functions is presented for stability and trajectory tracking of a quadrotor system.
Abstract: This paper presents a solution to stability and trajectory tracking of a quadrotor system using a model predictive controller designed using a type of orthonormal functions called Laguerre functions. A linear model of the quadrotor is derived and used. To check the performance of the controller we compare it with a linear quadratic regulator and a more traditional linear state space MPC. Simulations for trajectory tracking and stability are performed in MATLAB and results provided in this paper.
20 citations
TL;DR: In this article, a new method of backward differentiation formula (BDF-) type for solving fractional functional differential equations with delay (FDDEs) was proposed, based on the interval approximation of the true solution using the Clenshaw and Curtis formula that is based on truncated shifted Chebyshev polynomials.
Abstract: Fractional functional differential equations with delay (FDDEs) have recently played a significant role in modeling of many real areas of sciences such as physics, engineering, biology, medicine, and economics. FDDEs often cannot be solved analytically so the approximate and numerical methods should be adapted to solve these types of equations. In this paper we consider a new method of backward differentiation formula- (BDF-) type for solving FDDEs. This approach is based on the interval approximation of the true solution using the Clenshaw and Curtis formula that is based on the truncated shifted Chebyshev polynomials. It is shown that the new approach can be reformulated in an equivalent way as a Runge-Kutta method and the Butcher tableau of this method is given. Estimation of local and global truncating errors is deduced and this leads to the proof of the convergence for the proposed method. Illustrative examples of FDDEs are included to demonstrate the validity and applicability of the proposed approach.
20 citations
TL;DR: In this article, the authors have initiated the study of the best proximity point problem in the setup of generalized metric spaces, and some results dealing with existence and uniqueness of a coincidence best proximity points of mappings satisfying certain contractive conditions in such spaces are obtained.
Abstract: The aim of this paper is to initiate the study of coincidence best proximity point problem in the setup of generalized metric spaces. Some results dealing with existence and uniqueness of a coincidence best proximity point of mappings satisfying certain contractive conditions in such spaces are obtained. An example is provided to support the result proved herein. Our results generalize, extend, and unify various results in the existing literature.
TL;DR: The well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions.
Abstract: Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.
TL;DR: In this paper, the enhanced multistage homotopy perturbation method (EMHPM) was applied to solve delay differential equations (DDEs) with constant and variable coefficients, based on a sequence of subintervals that provide approximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithm solutions.
Abstract: We expand the application of the enhanced multistage homotopy perturbation method (EMHPM) to solve delay differential equations (DDEs) with constant and variable coefficients. This EMHPM is based on a sequence of subintervals that provide approximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithm solutions. To address the accuracy of our proposed approach, we examine the solutions of several DDEs having constant and variable coefficients, finding predictions with a good match relative to the corresponding numerical integration solutions.
TL;DR: In this article, generalized -contractive mappings of integral type in the context of generalized metric spaces were introduced and the results of this paper generalize and improve several results on the topic in literature.
Abstract: We introduce generalized -contractive mappings of integral type in the context of generalized metric spaces. The results of this paper generalize and improve several results on the topic in literature.
TL;DR: The reproducing kernel method with interpolation is used for finding approximate solutions of delay differential equations and the comparison of the results with exact ones is made to confirm the validity and efficiency.
Abstract: We use the reproducing kernel method (RKM) with interpolation for finding approximate solutions of delay differential equations. Interpolation for delay differential equations has not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency.
TL;DR: In this article, it was shown that the double inequality holds for all with if and only if the Sandor and th power means of and, respectively, are the same for all.
Abstract: We prove that the double inequality holds for all with if and only if and …, where and are the Sandor and th power means of and , respectively.
TL;DR: In this article, we established some inequalities of Simpson type involving Riemann-Liouville fractional integrals for mappings whose first derivatives are h-convex.
Abstract: We establish some inequalities of Simpson type involving Riemann-Liouville fractional integrals for mappings whose first derivatives are h-convex.
TL;DR: In this article, a new class of univalent harmonic functions is introduced and sufficient coefficient conditions for these classes are given if certain restrictions are imposed on the coefficients of these harmonic functions.
Abstract: New classes of univalent harmonic functions are introduced. We give sufficient coefficient conditions for these classes. These coefficient conditions are shown to be also necessary if certain restrictions are imposed on the coefficients of these harmonic functions. By using extreme points theory we also obtain coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. Radii of convexity and starlikeness of the classes are also considered.
TL;DR: In this paper, the determinants and inverses of the circulant type matrices are discussed and the invertibility of these matrices is shown, and the determinant and inverse matrices of these types are given, respectively.
Abstract: It is a hot topic that circulant type matrices are applied to networks engineering. The determinants and inverses of Tribonacci circulant type matrices are discussed in the paper. Firstly, Tribonacci circulant type matrices are defined. In addition, we show the invertibility of Tribonacci circulant matrix and present the determinant and the inverse matrix based on constructing the transformation matrices. By utilizing the relation between left circulant, -circulant matrices and circulant matrix, the invertibility of Tribonacci left circulant and Tribonacci -circulant matrices is also discussed. Finally, the determinants and inverse matrices of these matrices are given, respectively.
TL;DR: In this article, the authors presented explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematical physics and applied a theory of Frobenius decompositions to the coupled Burgers.
Abstract: Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematical physics are presented. By applying a theory of Frobenius decompositions and, more precisely, by using a transformation method to the coupled Burgers, combined Korteweg-de Vries- (KdV-) modified KdV and Schrodinger-KdV equation is written as bilinear ordinary differential equations and two solutions to describing nonlinear interaction of travelling waves are generated. The properties of the multiple travelling wave solutions are shown by some figures. All solutions are stable and have applications in physics.
TL;DR: In this paper, a novel derivation of a second-order accurate Grunwald-Letnikov-type approximation to the fractional derivative of a function is presented, under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation.
Abstract: A novel derivation of a second-order accurate Grunwald-Letnikov-type approximation to the fractional derivative of a function is presented. This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation. Some example functions are chosen and numerical results are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractional derivatives.
TL;DR: Analysis of the deterministic model with isolation and lost to follow-up of three strains of Mycobacterium tuberculosis indicates that the disease-free equilibrium is globally asymptotically stable (GAS) in the absence of disease reinfection.
Abstract: We present a deterministic model with isolation and lost to follow-up for the transmission dynamics of three strains of Mycobacterium tuberculosis (TB), namely, the drug sensitive, multi-drug-resistant (MDR), and extensively-drug-resistant (XDR) TB strains. The model is analyzed to gain insights into the qualitative features of its associated equilibria. Some of the theoretical and epidemiological findings indicate that the model has locally asymptotically stable (LAS) disease-free equilibrium when the associated reproduction number is less than unity. Furthermore, the model undergoes in the presence of disease reinfection the phenomenon of backward bifurcation, where the stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the associated reproduction number is less than unity. Further analysis of the model indicates that the disease-free equilibrium is globally asymptotically stable (GAS) in the absence of disease reinfection. The result of the global sensitivity analysis indicates that the dominant parameters are the disease progression rate, the recovery rate, the infectivity parameter, the isolation rate, the rate of lost to follow-up, and fraction of fast progression rates. Our results also show that increase in isolation rate leads to a decrease in the total number of individuals who are lost to follow-up.
TL;DR: The infinite matrix products are introduced including some of their main properties and convergence results and a limit representation of the matrix gamma function is provided.
Abstract: We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.
TL;DR: In this article, the authors formulated and solved the problem of relevant FRAP data selection, and analyzed and compared two approaches of data processing, i.e., the integrated data approach and the full (spatiotemporal) data approach.
Abstract: Fluorescence recovery after photobleaching (FRAP) is a widely used measurement technique to determine the mobility of fluorescent molecules within living cells. While the experimental setup and protocol for FRAP experiments are usually fixed, data (pre)processing represents an important issue. The aim of this paper is twofold. First, we formulate and solve the problem of relevant FRAP data selection. The theoretical findings are illustrated by the comparison of the results of parameter identification when the full data set was used and the case when the irrelevant data set (data with negligible impact on the confidence interval of the estimated parameters) was removed from the data space. Second, we analyze and compare two approaches of FRAP data processing. Our proposition, surprisingly for the FRAP community, claims that the data set represented by the FRAP recovery curves in form of a time series (integrated data approach commonly used by the FRAP community) leads to a larger confidence interval compared to the full (spatiotemporal) data approach.
TL;DR: Two classes of iterative methods are presented whose orders of convergence are four and five, respectively, for solving systems of nonlinear equations, by using the technique of weight functions in each step by adding only one functional evaluation of the vectorial nonlinear function.
Abstract: We present two classes of iterative methods whose orders of convergence are four and five, respectively, for solving systems of nonlinear equations, by using the technique of weight functions in each step. Moreover, we show an extension to higher order, adding only one functional evaluation of the vectorial nonlinear function. We perform numerical tests to compare the proposed methods with other schemes in the literature and test their effectiveness on specific nonlinear problems. Moreover, some real basins of attraction are analyzed in order to check the relation between the order of convergence and the set of convergent starting points.
TL;DR: In this article, the authors define a new class of -analogue of -valently closed-to-convex functions, and introduce new class by means of this new general differential operator.
Abstract: By making use of the concept of fractional -calculus, we firstly define -extension of the generalization of the generalized Al-Oboudi differential operator. Then, we introduce new class of -analogue of -valently closed-to-convex function, and, consequently, new class by means of this new general differential operator. Our main purpose is to determine the general properties on such class and geometric properties for functions belonging to this class with negative coefficient. Further, the -extension of interesting properties, such as distortion inequalities, inclusion relations, extreme points, radii of generalized starlikeness, convexity and close-to-convexity, quasi-Hadamard properties, and invariant properties, is obtained. Finally, we briefly indicate the relevant connections of our presented results to the former results.
TL;DR: In this article, the authors study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space.
Abstract: In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature.
TL;DR: In this article, the authors investigated the unsteady peristaltic transport of a viscoelastic fluid with fractional Maxwell model through two coaxial vertical tubes and obtained an analytical solution of the problem by using a fractional calculus approach.
Abstract: We investigate the unsteady peristaltic transport of a viscoelastic fluid with fractional Maxwell model
through two coaxial vertical tubes. This analysis has been carried under low Reynolds number and long wavelength approximations.
Analytical solution of the problem is obtained by using a fractional calculus approach. The effects of Grashof number, heat
parameter, relaxation time, fractional time derivative parameter, amplitude ratio, and the radius ratio on the pressure gradient,
pressure rise, and the friction forces on the inner and outer tubes are graphically presented and discussed.
TL;DR: In this paper, the authors studied the obstacle problem for second order nonlinear equations whose model appears in the stationary diffusion-convection problem and assumed that the growth coefficient of the convection term lies in the Marcinkiewicz space.
Abstract: We study the obstacle problem for second order nonlinear equations whose model appears in the stationary diffusion-convection problem. We assume that the growth coefficient of the convection term lies in the Marcinkiewicz space -.
TL;DR: In this paper, the authors introduce two subclasses of meromorphic functions and investigate convolution properties, coefficient estimates, and containment properties for these subclasses for functions of the form, which are analytic in the punctured unit disc.
Abstract: Making use of the operator for functions of the form , which are analytic in the punctured unit disc and , we introduce two subclasses of meromorphic functions and investigate convolution properties, coefficient estimates, and containment properties for these subclasses.