Volume 24, Issue 2

This paper addresses the numerical solution of multi-dimensional variable-order fractional Gross–Pitaevskii equations (VOF-GPEs) with initial and boundary conditions. A new scheme is proposed based on the fully shifted fractional Jacobi collocation method and adopting two independent approaches: (i) the discretization of the space variable and (ii) the discretization of the time variable. A complete theoretical formulation is presented and numerical examples are given to illustrate the performance and efficiency of the new algorithm. The superiority of the scheme to tackle VOF-GPEs is revealed, even when dealing with nonsmooth time solutions.

In this paper, the dynamical behaviors of a multiple delayed HIV-1 infection model which describes the interactions of humoral, cytotoxic T lymphocyte (CTL) immune responses, and two modes of transmission that are the classical virus-to-cell infection and the direct cell-to-cell transmission are investigated. The model incorporates three delays, including the delays of cell infection, virus production and activation of immune response. We first prove the well-posedness of the model, and calculate the biological existence of equilibria and the reproduction numbers, which contain virus infection, humoral immune response, CTL immune response, CTL immune competition, and humoral immune competition. Further, the threshold conditions for the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response, and interior are established by utilizing linearization method and the Lyapunov functionals. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics.

In this study, we give the numerical scheme to the stochastic nonlinear advection diffusion equation. This models is considered with white noise (or random process) having same intensity by changing frequencies. Furthermore, the stability and consistency of proposed scheme are also discussed. Moreover, it is concerned about the analytical solutions, the Riccati equation mapping method is adopted. The different families of single (shock and singular) and mixed (complex solitary-shock, shock-singular, and double-singular) form solutions are obtained with the different choices of free parameters. The graphical behavior of solutions is also depicted in 3D and corresponding contours.

Multistaged crystallization systems are used in the production of many chemicals. In this article, employing the population balance framework, we develop a model for a column crystallizer where particle agglomeration is a significant growth mechanism. The main part of the model can be reduced to a system of integrodifferential equations (IDEs) of the Volterra type. To solve this system simultaneously, we examine two numerical schemes that yield a direct method of solution and an implicit Runge–Kutta type method. Our numerical experiments show that the extension of a Hermite predictor–corrector method originally advanced in Khanh (1994) for a single IDE is effective in solving our model. The numerical method is presented for a generalization of the model which can be used to study and simulate a number of possible operating profiles of the column.

The aim of this paper is to derive a novel discrete form of stochastic fractional Grönwall lemma involving a martingale. The proof of the derived inequality is accomplished by a corresponding no randomness form of the discrete fractional Grönwall inequality and an upper bound for discrete-time martingales representing the supremum in terms of the infimum. The release of a martingale term on the right-hand side of the given inequality and the graded L 1 difference formula for the time Caputo fractional derivative of order 0 < α < 1 on the left-hand side are the main challenges of the stated and proved main theorem. As an example of application, the constructed theorem is used to derive an a priori estimate for a discrete stochastic fractional model at the end of the paper.

The purpose of this paper is to study a new Tseng’s extragradient method with two different stepsize rules for solving pseudomonotone variational inequalities in real Hilbert spaces. We prove a strong convergence theorem of the proposed algorithm under some suitable conditions imposed on the parameters. Moreover, we also give some numerical experiments to demonstrate the performance of our algorithm.

Integration using least squares method in space and Crank–Nicolson approach in time is managed to set up an algorithm to solve the RLW equation numerically. Trial functions in the least square method consist of a combination of the quartic B-spline functions. Integration of the RLW equation gives a system of algebraic equations. The solutions consisting of a combination of the quartic B-splines are given for some initial and boundary value problems of RLW equation.

This paper concerns with the integrability of variable coefficient fifth order nonlinear Schrödinger’s equation describing the dynamics of attosecond pulses in inhomogeneous fibers. Variable coefficients incorporate varying dispersion and nonlinearity which are of physical significance in considering the nonuniform boundaries of fibers as well as the inhomogeneities of the media. The well-known exp(− φ ( s ))-expansion method is used to retrieve singular and periodic solitons with the aid of symbolic computation. The structures of the obtained solutions are discussed along with their existence criteria. Moreover, the modulation instability analysis is carried out to identify the instability regions. A dispersion relation is extracted between wave number and frequency. The optimal value of the frequency is found for the occurrence of the instability. A detailed discussion of the results is also given along with graphics.

This paper is mainly concerned with some new asymptotic properties on mild solutions to a nonlocal Cauchy problem of integrodifferential equation in Banach spaces. Under some well-imposed conditions on the nonlocal Cauchy, the neutral and forced terms, respectively, we establish some existence results for weighted pseudo S -asymptotically ( ω , k )-Bloch periodic mild solutions to the referenced equation on R + ${\mathbb{R}}_{+}$ by suitable superposition theorems. The results show that the strict contraction of the nonlocal Cauchy and the neutral terms with the state variable has an appreciable effect on the existence and uniqueness of such a solution compared with the forced term. As an auxiliary result, the existence of weighted pseudo S -asymptotically ( ω , k )-Bloch periodic mild solutions is deduced under the sublinear growth condition on the force term with its state variable. The existence of weighted pseudo S -asymptotically ω -antiperiodic mild solution is also obtained as a special example.

The subgradient extragradient method with inertial extrapolation step x n + θ n ( x n − x n −1 ) (also known as inertial subgradient extragradient method) has been studied extensively in the literature for solving variational inequalities and equilibrium problems. Most of the inertial subgradient extragradient methods in the literature for both variational inequalities and equilibrium problems have not considered the special case when the inertial factor θ n = 1. The convergence results have always been obtained when the inertial factor θ n is assumed 0 ≤ θ n < 1. This paper considers the relaxed inertial version of subgradient extragradient method for equilibrium problems with 0 ≤ θ n ≤ 1. We give both weak and strong convergence results using this inertial subgradient extragradient method and also give some numerical illustrations.

The improved tan( φ /2)-expansion, simplest equation, and extended (G′/G)-expansion methods are employed to construct the exact solutions involving parameters of the Van der Waals equation arising in the material industry. This model explains the phase separation phenomenon. Understanding the prominent dynamic and static properties of this model and other models of this type is of great importance for the physical phenomena encountered in many areas of industry. Therefore, for such models, it is also important to obtain guiding solutions in obtaining new information. Many explicit wave solutions consisting of trigonometric, hyperbolic, rational, and exponential functions are found by using analytical techniques. The obtained solutions were verified with Maple by placing them back into the original equations. Moreover, graphical demonstrations for some of the obtained solutions are given.

This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ -Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

The present research work is basically devoted to construction of a fractional order differential equation with time delay. Initially, integral representation is given to solution of the underline problem. Afterwards, operator form of solution is studied under some auxiliary hypothesis. Since uniqueness of solution is required, therefore we also provide results for exploring the uniqueness of solution for the underlying model. Using Lebesgue dominated convergence theorem and some other results from analysis, this work provides results devoted to existence of at least one solution. Also, for investigating the nature of solution for the proposed model, we study different kind of stability analysis. These stability related results show, how the solution behave with time. At the end of the article, we illustrate the obtained results via some examples.

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

In this study, we dealt with the new conservation theorem and the auxiliary method. We have applied the theorem and method to (3 + 1)-dimensional modified Wazwaz–KdV equations. When we applied a new conservation theorem to given equations, the obtained conservation laws did not satisfy the divergence condition. So, we modified the obtained conservation laws. These conservation laws contain extra terms. Finally, we applied the auxiliary method to given equations. We obtained two solution families with six exact solutions. All the obtained solutions are different from each other. For a suitable value of the solutions, the 3D and 2D surfaces have been plotted by Maple.

In this article, we focus on stability and ψ -algebraic decay (algebraic decay in the sense of ψ -function) of the equilibrium to the nonlinear ψ -fractional ordinary differential system. Before studying the nonlinear case, we show the stability and decay for linear system in more detail. Then we establish the linearization theorem for the nonlinear system near the equilibrium and further determine the stability and decay rate of the equilibrium. Such discussions include two cases, one with ψ -Caputo fractional derivative, another with ψ -Riemann–Liouville derivative, where the latter is a bit more complex than the former. Besides, the integral transforms are also provided for future studies.

In our paper, we computed some new characterizations due to an alternative modified frame N ⃗ , C ⃗ , W ⃗ $\left\{ \overrightarrow {N}, \overrightarrow {C}, \overrightarrow {W}\right\}$ in Euclidean 3-space and we get general differential equation characterizations of a space curve due to the vectors N ⃗ , C ⃗ , W ⃗ $ \overrightarrow {N}, \overrightarrow {C}, \overrightarrow {W}$ . Furthermore, we investigated some differential equations characterizations of the harmonic and harmonic 1-type curves.

The aim of the work is to describe the air flow in an enclosed space, which is ventilated by a diffuser, to select an appropriate turbulence model, to solve the problem using the ANSYS Fluent, to study the effect of heat sources in a room on air flow under various conditions and to simulate the movement of particulate matter. As a result, the distribution of PM2.5 particles in the room was shown, which enter the room through the diffuser. According to the data obtained, the temperature value increases with an increase in the area of the heat source, that is, with an increase in the number of batteries. The maximum temperature corresponds to a room with a warm floor, the minimum temperature is observed in a room with one battery. The obtained numerical data can be used when installing ventilation or heating devices inside buildings, when simulating the movement of harmful particles in the air, when determining the optimal ways to clean the air.

In this paper, we propose two inertial accelerated algorithms which do not require prior knowledge of operator norm for solving split feasibility problem with multiple output sets in real Hilbert spaces. We prove weak and strong convergence results for approximating the solution of the considered problem under certain mild conditions. We also give some numerical examples to demonstrate the performance and efficiency of our proposed algorithms over some existing related algorithms in the literature.

In this manuscript, several types of exact solutions including trigonometric, hyperbolic, exponential, and rational function are successfully constructed with the implementation of two modified mathematical methods, namely called extended simple equation and modified F-expansion methods on the (2 + 1)-dimensional Date–Jimbo–Kashiwara–Miwa and the combined sinh–cosh-Gordon equations. Diverse form of solitary wave solutions is achieved from exact solutions by passing the special values to the parameters. Some solution are plotted in the form of 3D and 2D by assigning the specific values to parameters under the constrain condition to the solutions. These approaches yield the new solutions that we think other researchers have missed in the field of nonlinear sciences. Hence the searched wave’s results are loyal to the researchers and also have imperious applications in applied sciences.

Refugia have been perceived as a major role in structuring species biodiversity, and understanding the impacts of this force in a community assembly with prey–predator species is a difficult task because refuge process can interact with different ecological components and may show counterintuitive effects. To understand this problem, we used a simple two-species model incorporating a functional response inspired by a Holling type-II equation and a prey refuge mechanism that depends on prey and predator population densities (i.e., density-dependent prey refuge). We then perform the co-dimension one and co-dimension two bifurcation analysis to examine steady states and its stability, together with the bifurcation points as different parameters change. As the capacity of prey refuge is varied, there occur critical values i.e., saddle-node and supercritical Hopf bifurcations. The interaction between these two co-dimension one bifurcations engenders distinct outcomes of ecological system such as coexistence of species, bistability phenomena and oscillatory dynamics. Additionally, we construct a parameter space diagram illustrating the dynamics of species interactions as prey refuge intensity and predation pressure vary; as the two saddle-node move nearer to one another, these bifurcations annihilate tangentially in a co-dimension two cusp bifurcation. We also realised several contrasting observations of refuge process on species biodiversity: for instance, while it is believed that some refuge processes (e.g., constant proportion of prey refuge) would result in exclusion of predator species, our findings show that density-dependent prey refuge is beneficial for both predator and prey species, and consequently, promotes the maintenance of species biodiversity.