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In this paper, we consider the inverse problem of derivative order estimation in a fractional logistic model. In order to solve the direct problem, we use the Gr&uuml, nwald-Letnikov fractional derivative, then the inverse problem is tackled within a Bayesian perspective. To construct the likelihood function, we propose an explicit numerical scheme based on the truncated series of the derivative definition. By MCMC samples of the marginal posterior distributions, we estimate the order of the derivative and the growth rate parameter in the dynamic model, as well as the noise in the observations. To evaluate the methodology, a simulation was performed using synthetic data, where the bias and mean square error are calculated, the results give evidence of the effectiveness for the method and the suitable performance of the proposed model. Moreover, an example with real data is presented as evidence of the relevance of using a fractional model.

A generalized form of the Mittag-Leffler function denoted by p E q, &delta, &lambda, &mu, &nu, x is established and studied in this paper. The fractional integrals involving the newly defined function are investigated. As an application, the solutions of a generalized fractional kinetic equation containing this function are derived and the nature of the solution is studied with the help of graphical analysis.

The purpose of this paper is to present some topological properties in E-metric spaces such as the properties of e-sequences, the decision conditions of e-Cauchy sequences, the characteristics of non-normal cones, and so on. Moreover, the theorem of nested closed-balls in such spaces is displayed. In addition, some principal applications to fixed point theory are also given.

In this paper, we present some new generalizations of Mizoguchi-Takahashi&rsquo, s fixed point theorem which also improve and extend Du-Hung&rsquo, s fixed point theorem. Some new examples illustrating our results are also given. By applying our new results, some new fixed point theorems for essential distances and e 0 -metrics were established.

In this paper we consider ( s − q ) -graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results.

In this paper, we are interested in the pseudomonotone variational inequalities and fixed point problem of pseudocontractive operators in Hilbert spaces. An iterative algorithm has been constructed for finding a common solution of the pseudomonotone variational inequalities and fixed point of pseudocontractive operators. Strong convergence analysis of the proposed procedure is given. Several related corollaries are included.

In the paper, we study the oscillatory and asymptotic properties of solutions to a class of third-order linear neutral delay differential equations with noncanonical operators. Via the application of comparison principles with associated first and second-order delay differential inequalities, we offer new criteria for the oscillation of all solutions to a given differential equation. Our technique essentially simplifies the process of investigation and reduces the number of conditions required in previously known results. The strength of the newly obtained results is illustrated on the Euler type equations.

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space.

In this paper, we study Linear Riemann-Liouville fractional differential equations with a constant delay. The initial condition is set up similarly to the case of ordinary derivative. Explicit formulas for the solutions are obtained for various initial functions.

Surface modeling is closely related to interpolation and approximation by using level set methods, radial basis functions methods, and moving least squares methods. Although radial basis functions with global support have a very good approximation effect, this is often accompanied by an ill-conditioned algebraic system. The exceedingly large condition number of the discrete matrix makes the numerical calculation time consuming. The paper introduces a truncated exponential function, which is radial on arbitrary n-dimensional space R n and has compact support. The truncated exponential radial function is proven strictly positive definite on R n while internal parameter l satisfies l &ge, &lfloor, n 2 &rfloor, + 1 . The error estimates for scattered data interpolation are obtained via the native space approach. To confirm the efficiency of the truncated exponential radial function approximation, the single level interpolation and multilevel interpolation are used for surface modeling, respectively.

In this paper, a quantum trapezium-type inequality using a new class of function, the so-called generalized ϕ -convex function, is presented. A new quantum trapezium-type inequality for the product of two generalized ϕ -convex functions is provided. The authors also prove an identity for twice q - differentiable functions using Raina&rsquo, s function. Utilizing the identity established, certain quantum estimated inequalities for the above class are developed. Various special cases have been studied. A brief conclusion is also given.

In this paper, we consider and study a system of multi-valued mixed variational inclusions with XOR-operation &oplus, in real ordered uniformly smooth Banach spaces. This system consists of bimappings, multi-valued mappings and Cayley operators. An iterative algorithm is suggested to find the solution to a system of multi-valued mixed variational inclusions with XOR-operation &oplus, and consequently an existence and convergence result is proved. In support of our main result, an example is constructed.

Recently, the partially degenerate Bell polynomials and numbers, which are a degenerate version of Bell polynomials and numbers, were introduced. In this paper, we consider the new type degenerate Bell polynomials and numbers, and obtain several expressions and identities on those polynomials and numbers. In more detail, we obtain an expression involving the Stirling numbers of the second kind and the generalized falling factorial sequences, Dobinski type formulas, an expression connected with the Stirling numbers of the first and second kinds, and an expression involving the Stirling polynomials of the second kind.

In this paper, we introduce and prove several generalized algebraic-trigonometric inequalities by considering negative exponents in the inequalities.

Foeplitz and Loeplitz matrices are Toeplitz matrices with entries being Fibonacci and Lucas numbers, respectively. In this paper, explicit expressions of determinants and inverse matrices of Foeplitz and Loeplitz matrices are studied. Specifically, the determinant of the n &times, n Foeplitz matrix is the ( n + 1 ) th Fibonacci number, while the inverse matrix of the n &times, n Foeplitz matrix is sparse and can be expressed by the nth and the ( n + 1 ) th Fibonacci number. Similarly, the determinant of the n &times, n Loeplitz matrix can be expressed by use of the ( n + 1 ) th Lucas number, and the inverse matrix of the n &times, n ( n &gt, 3 ) Loeplitz matrix can be expressed by only seven elements with each element being the explicit expressions of Lucas numbers. Finally, several numerical examples are illustrated to show the effectiveness of our new theoretical results.

In this paper, we introduce the iterative scheme for finding a common fixed point of a countable family of G-nonexpansive mappings by the shrinking projection method which generalizes Takahashi Takeuchi and Kubota&rsquo, s theorem in a Hilbert space with a directed graph. Simultaneously, we give examples and numerical results for supporting our main theorems and compare the rate of convergence of some examples under the same conditions.

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L &infin, error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

The purpose of this paper is to introduce an iterative algorithm of two sequences which depend on each other by using the intermixed method. Then, we prove a strong convergence theorem for solving fixed-point problems of nonlinear mappings and we treat two variational inequality problems which form an approximate modified generalized system of variational inequalities (MGSV). By using our main theorem, we obtain the additional results involving the split feasibility problem and the constrained convex minimization problem. In support of our main result, a numerical example is also presented.

This paper explores a certain relationship between the almost fixed point property (AFPP for short) of a compact and n-dimensional Euclidean space and that of its digitized space. Based on several types of digitizations, we prove that the AFPP of a compact and n-dimensional Euclidean cube is preserved by each of the U ( k ) , the L ( k ) and the Khalimsky digitizations, k &isin, 3 n - 1 , n &isin, N .

In this paper, we present a new generalization of the Perov fixed point theorem on vector-valued metric space. Moreover, to show the significance of our result, we present both a nontrivial comparative example and an application to a kind of semilinear operator system about the existence of its solution.

In this paper, we introduce the concept of coincidence best proximity point for multivalued Suzuki-type &alpha, admissible mapping using &theta, contraction in b-metric space. Some examples are presented here to understand the use of the main results and to support the results proved herein. The obtained results extend and generalize various existing results in literature.

The aim of this paper is to introduce an orignal coupling procedure between surface integral equation formulations and on-surface radiation condition (OSRC) methods for solving two-dimensional scattering problems for non convex structures. The key point is that the use of the OSRC introduces a sparse block in the surface operator representation of the wave field while the integral part leads to an improved accuracy of the OSRC method in the non convex part of the scattering structure. The procedure is given for both the Dirichlet and Neumann scattering problems. Some numerical simulations show the improvement induced by the coupling method.

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

In this paper, a new efficient and practical modification of the Adomian decomposition method is proposed with Laguerre polynomials and the second kind of Chebyshev polynomials which has not been introduced in other articles to the best of our knowledge. This approach can be utilized to approximately solve linear and nonlinear differential equations. The proposed formulations are examined by a representative example and the numerical results confirm their efficiency and accuracy.

In recent years, predator–prey systems have increased their applications and have given rise to systems which represent more accurately different biological issues that appear in the context of interacting species. Our aim in this paper is to give a state-of-the-art review of recent predator–prey models which include some interesting characteristics such as Allee effect, fear effect, cannibalism, and immigration. We compare the qualitative results obtained for each of them, particularly regarding the equilibria, local and global stability, and the existence of limit cycles.

A class of p-valent analytic functions is introduced using the q-difference operator and the familiar Janowski functions. Several properties of functions in the class, such as the Fekete–Szegö inequality, coefficient estimates, necessary and sufficient conditions, distortion and growth theorems, radii of convexity and starlikeness, closure theorems and partial sums, are discussed in this paper.

Partial difference equations have received more and more attention in recent years due to their extensive applications in diverse areas. In this paper, we consider a Dirichlet boundary value problem of the partial difference equation involving the mean curvature operator. By applying critical point theory, the existence of at least three solutions is obtained. Furthermore, under some appropriate assumptions on the nonlinearity, we respectively show that this problem admits at least two or three positive solutions by means of a strong maximum principle. Finally, we present two concrete examples and combine with images to illustrate our main results.

In this paper, the method of artificial boundary conditions for exterior quasilinear problems in concave angle domains is investigated. Based on the Kirchhoff transformation, the exact quasiliner elliptical arc artificial boundary condition is derived. Using the approximate elliptical arc artificial boundary condition, the finite element method is formulated in a bounded region. The error estimates are obtained. The effectiveness of our method is showed by some numerical experiments.

In this paper, we introduce the notion of bicomplex partial metric space and prove some common fixed point theorems. The presented results generalize and expand some of the literature’s well-known results. An example and application on bicomplex partial metric space is given.

This paper established a new high precision Wilker-type inequality with fractional powers for the function 2−[x/sinx6/5+x/tanx3/5] bounded by the function x6tanx/x5/4.

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

In this paper, we give sufficient conditions for the existence and uniqueness of the solution of Sturm–Liouville equations subject to Dirichlet boundary value conditions and involving Kurzweil–Henstock integrable functions on unbounded intervals. We also present a finite element method scheme for Kurzweil–Henstock integrable functions.

An operator-based scheme for the numerical integration of fractional differential equations is presented in this paper. The generalized differential operator is used to construct the analytic solution to the corresponding characteristic ordinary differential equation in the form of an infinite power series. The approximate numerical solution is constructed by truncating the power series, and by changing the point of the expansion. The developed adaptive integration step selection strategy is based on the controlled error of approximation induced by the truncation. Computational experiments are used to demonstrate the efficacy of the proposed scheme.

For a discrete dynamical system, the prime orbit and Mertens’ orbit counting functions describe the growth of its closed orbits in a certain way. The asymptotic behaviours of these counting functions can be determined via Artin–Mazur zeta function of the system. Specifically, the existence of a non-vanishing meromorphic extension of the zeta function leads to certain asymptotic results. In this paper, we prove the asymptotic behaviours of the counting functions for a certain type of shift spaces induced by directed bouquet graphs and Dyck shifts. We call these shift spaces as the bouquet-Dyck shifts. Since their respective zeta function involves square roots of polynomials, the meromorphic extension is difficult to be obtained. To overcome this obstacle, we employ some theories on zeros of polynomials, including the well-known Eneström–Kakeya Theorem in complex analysis. Finally, the meromorphic extension will imply the desired asymptotic results.

In the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very “toxic”, in the sense that it may force the finite precision error accumulation to grow arbitrarily large, under specific conditions that are fully described here. First, an ensemble of definitions of general applicability is given for the rigorous determination of the number of erroneous digits accumulated in any quantity of an arbitrary algorithm. Next, the exact number of erroneous digits produced in a single multiplication is given as a function of the involved operands, together with formulae offering the corresponding probabilities. In case the statistical properties of these operands are known, exact evaluation of the aforementioned probabilities takes place. Subsequently, the statistical properties of the accumulated finite precision error during any number of successive multiplications are explicitly analyzed. A method for exact tracking of this accumulated error is presented, together with associated theorems. Moreover, numerous dedicated experiments are developed and the corresponding results that fully support the theoretical analysis are given. Eventually, a number of important, probable and possible applications is proposed, where all of them are based on the methodology and the results introduced in the present work. The proposed methodology is expandable, so as to tackle the round-off error analysis in all arithmetic operations.

In this paper, for solving the nonlinear Rosenau-KdV equation, a conservative implicit two-level nonlinear scheme is proposed by a new numerical method named the multiple integral finite volume method. According to the order of the original differential equation’s highest derivative, we can confirm the number of integration steps, which is just called multiple integration. By multiple integration, a partial differential equation can be converted into a pure integral equation. This is very important because we can effectively avoid the large errors caused by directly approximating the derivative of the original differential equation using the finite difference method. We use the multiple integral finite volume method in the spatial direction and use finite difference in the time direction to construct the numerical scheme. The precision of this scheme is O(τ2+h3). In addition, we verify that the scheme possesses the conservative property on the original equation. The solvability, uniqueness, convergence, and unconditional stability of this scheme are also demonstrated. The numerical results show that this method can obtain highly accurate solutions. Further, the tendency of the numerical results is consistent with the tendency of the analytical results. This shows that the discrete scheme is effective.

Developing new and efficient numerical integration techniques is of great importance in applied mathematics and computer science. Among the variety of available methods, multistep ODE solvers are broadly used in simulation software. Recently, semi-implicit integration proved to be an efficient compromise between implicit and explicit ODE solvers, and multiple high-performance semi-implicit methods were proposed. However, the computational efficiency of any ODE solver can be significantly increased through the introduction of an adaptive integration stepsize, but it requires the estimation of local truncation error. It is known that recently proposed extrapolation semi-implicit multistep methods (ESIMM) cannot operate with existing local truncation error (LTE) estimators, e.g., embedded methods approach, due to their specific right-hand side calculation algorithm. In this paper, we propose two different techniques for local truncation error estimation and study the performance of ESIMM methods with adaptive stepsize control. The first considered approach is based on two parallel semi-implicit solutions with different commutation orders. The second estimator, called the “double extrapolation” method, is a modification of the embedded method approach. The introduction of the double extrapolation LTE estimator allowed us to additionally increase the precision of the ESIMM solver. Using several known nonlinear systems, including stiff van der Pol oscillator, as the testbench, we explicitly show that ESIMM solvers can outperform both implicit and explicit linear multistep methods when implemented with an adaptive stepsize.

The Gray–Scott (GS) model is a non-linear system of equations generally adopted to describe reaction–diffusion dynamics. In this paper, we discuss a numerical scheme for solving the GS system. The diffusion coefficients of the model are on surfaces and they depend on space and time. In this regard, we first adopt an implicit difference stepping method to semi-discretize the model in the time direction. Then, we implement a hybrid advanced meshless method for model discretization. In this way, we solve the GS problem with a radial basis function–finite difference (RBF-FD) algorithm combined with the closest point method (CPM). Moreover, we design a predictor–corrector algorithm to deal with the non-linear terms of this dynamic. In a practical example, we show the spot and stripe patterns with a given initial condition. Finally, we experimentally prove that the presented method provides benefits in terms of accuracy and performance for the GS system’s numerical solution.

Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results.

The aim of this paper is to generalize the F -contractive condition in the framework of &alpha, &minus, &nu, complete modular b-metric spaces. Some results in ordered modular b-metric spaces are also presented. Moreover, an illustrative example and some related applications are presented to support the obtained results.

Recently, George et al. (in Georgea, R., Radenovicb, S., Reshmac, K.P., Shuklad, S. Rectangular b-metric space and contraction principles. J. Nonlinear Sci. Appl. 2015, 8, 1005&ndash, 1013) furnished the notion of rectangular b-metric pace (RBMS) by taking the place of the binary sum of triangular inequality in the definition of a b-metric space ternary sum and proved some results for Banach and Kannan contractions in such space. In this paper, we achieved fixed-point results for a pair of F-dominated mappings fulfilling a generalized rational F-dominated contractive condition in the better framework of complete rectangular b-metric spaces complete rectangular b-metric spaces. Some new fixed-point results with graphic contractions for a pair of graph-dominated mappings on rectangular b-metric space have been obtained. Some examples are given to illustrate our conclusions. New results in ordered spaces, partial b-metric space, dislocated metric space, dislocated b-metric space, partial metric space, b-metric space, rectangular metric spaces, and metric space can be obtained as corollaries of our results.

This paper is devoted to studying the existence of solutions to a class of impulsive fractional differential equations with derivative dependence. The used technical approach is based on variational methods and iterative methods. In addition, an example is given to demonstrate the main results.

In this paper, using an interpolative approach, we investigate two fixed point theorems in the framework of a b-metric space whose all closed and bounded subsets are compact. One of the theorems is for set-valued Hardy&ndash, Rogers-type and the other one is for set-valued Reich&ndash, Rus&ndash, Ćirić-type contractions. Examples are provided to validate the results.

The generalized time fractional Kolmogorov&ndash, Petrovsky&ndash, Piskunov equation (FKPP), D t &alpha, &omega, ( x , t ) = a ( x , t ) D x x &omega, ( x , t ) + F ( &omega, ( x , t ) ) , which plays an important role in engineering, chemical reaction problem is proposed by Caputo fractional order derivative sense. In this paper, we develop a framework wavelet, including shift Chebyshev polynomial of the first kind as a mother wavelet, and also construct some operational matrices that represent Caputo fractional derivative to obtain analytical solutions for FKPP equation with three different types of Initial Boundary conditions (Dirichlet, Dirichlet&ndash, Neumann, and Neumann&ndash, Robin). Our results shown that the Chebyshev wavelet is a powerful method, due to its simplicity, efficiency in analytical approximations, and its fast convergence. The comparison of the Chebyshev wavelet results indicates that the proposed method not only gives satisfactory results but also do not need large amount of CPU times.

This paper is devoted to an analysis on locating and counting satellite components born along the stability circle in the parameter space for a family of Jarratt-like iterative methods. An elementary theory of plane geometric curves is pursued to locate bifurcation points of such satellite components. In addition, the theory of Farey sequence is adopted to count the number of the satellite components as well as to characterize relationships between the bifurcation points. A linear stability theory on local bifurcations is developed based upon a small perturbation about the fixed point of the iterative map with a control parameter. Some properties of fixed and critical points under the M&ouml, bius conjugacy map are investigated. Theories and examples on locating and counting bifurcation points of satellite components in the parameter space are presented to analyze the bifurcation behavior underlying the dynamics behind the iterative map.

In this paper, we introduce the new concepts of K-adjustability convexity and strictly K-adjustability convexity which respectively generalize and extend the concepts of K-convexity and strictly K-convexity. We establish some new existence and uniqueness theorems of zeros for vector-valued functions with K-adjustability convexity. As their applications, we obtain existence theorems for the minimization problem and fixed point problem which are original and quite different from the known results in the existing literature.

This paper is devoted to investigating a class of nonhomogeneous Choquard equations with perturbation involving p-Laplacian. Under suitable hypotheses about the perturbation term, the existence of at least two nontrivial solutions for the given problems is obtained using Nehari manifold and minimax methods.

In the present paper, we investigate some Hermite-Hadamard ( HH ) inequalities related to generalized Riemann-Liouville fractional integral ( GRLFI ) via exponentially convex functions. We also show the fundamental identity for GRLFI having the first order derivative of a given exponentially convex function. Monotonicity and exponentially convexity of functions are used with some traditional and forthright inequalities. In the application part, we give examples and new inequalities for the special means.

This paper is concerned with proving the existence of solutions for a coupled system of quadratic integral equations of fractional order in Banach algebras. This result is a direct application of a fixed point theorem of Banach algebras. Some particular cases, examples and remarks are illustrated. Finally, the stability of solutions for that coupled system are studied.

In this paper, we propose a new global optimization algorithm, which can better solve a class of linear fractional programming problems on a large scale. First, the original problem is equivalent to a nonlinear programming problem: It introduces p auxiliary variables. At the same time, p new nonlinear equality constraints are added to the original problem. By classifying the coefficient symbols of all linear functions in the objective function of the original problem, four sets are obtained, which are I i + , I i − , J i + and J i − . Combined with the multiplication rule of real number operation, the objective function and constraint conditions of the equivalent problem are linearized into a lower bound linear relaxation programming problem. Our lower bound determination method only needs e i T x + f i ≠ 0 , and there is no need to convert molecules to non-negative forms in advance for some special problems. A output-space branch and bound algorithm based on solving the linear programming problem is proposed and the convergence of the algorithm is proved. Finally, in order to illustrate the feasibility and effectiveness of the algorithm, we have done a series of numerical experiments, and show the advantages and disadvantages of our algorithm by the numerical results.