Math Comput Model Research Articles

Approximation solution of the squeezing flow by the modification of optimal homotopy asymptotic method

The approximate solution of the squeezing axisymmetric fluid flow between infinite plates is discussed in the present paper. An optimal homotopy analysis method with a modification function technique is proposed to solve a class of nonlocal boundary value problems, namely squeezing axisymmetric fluid flow equation. We first transform the nonlocal boundary value problems into an equivalent integral equation, and then the optimal homotopy analysis technique is utilized for an approximate solution. The numerical results confirm the reliability of the present method as it tackles such nonlocal problems without any limiting assumptions. The proposed method is tested upon squeezing axisymmetric fluid flow equation from the literature and the results are compared with the available approximate solutions including perturbation method (Ran et al. in Commun Nonlinear Sci Numer Simul, 2007. https://doi.org/10.1016/j.cnsns ), homotopy perturbation method (Ran et al. 2007), homotopy analysis method (Ran et al. 2007), and optimal homotopy asymptotic method (Idrees et al. in Math Comput Model 55:1324–1333, 2012). The convergence and error analysis of the proposed method is discussed. It can be said that squeezing the axisymmetric fluid flow equation exists in different dynamical behaviors. In addition, the physical behaviors of these new exact solutions are given with two and three-dimensional graphs.

Some Inequalities for the Coefficients in Generalized Fourier Expansions

In this paper, we derive inequalities for the coefficients in generalized Fourier expansions of (m, n) convex functions in the sense of Popoviciu. Classical Fourier expansions as well as expansions relative to orthogonal polynomials are considered. The results presented here generalize the ones obtained by Niculescu and Rovenţa (Positivity 24(1):129–139, 2020). Some of the results obtained in deriving inequalities for these coefficients can be further used in obtaining Favard-type inequalities similar to the ones given in Wulbert (Math Comput Model 37(12–13):1383–1391, 2003). Favard type inequalities can be used in obtaining probabilistic inequalities which may be further used in fields such as statistical machine learning.

Improved algorithms for ranking and unranking (k, m)-ary trees in B-order

Du and Liu (Eur J Comb 28:1312–1321, 2007) introduced (k, m)-ary trees as a generalization of k-ary trees. In a (k, m)-ary tree, every node on even level has degree k (i.e., has k children), and every node on odd level has degree m (which is called a crucial node) or is a leaf. In particular, a (k, m)-ary tree of order n has exactly n crucial nodes. Recently, Amani and Nowzari-Dalini (Bull Iranian Math Soc 45(4):1145–1158, 2019) presented a generation algorithm to produce all (k, m)-ary trees of order n in B-order using Zaks’ encoding, and showed that the generated ordering of this encoding results in a reverse-lexicographical ordering. They also proposed the corresponding ranking and unranking algorithms for (k, m)-ary trees according to such a generated ordering. These algorithms take $$\mathcal {O}(kmn^2)$$ time and space for building a precomputed table in which (k, m)-Catalan numbers (i.e., a kind of generalized Catalan numbers) are stored in advance. Then, each ranking and unranking algorithm can be performed subsequently in $$\mathcal {O}(n)$$ and $$\mathcal {O}(n\log n)$$ time, respectively. In this paper, we revisit the ranking and unranking problems. With the help of an encoding scheme called “right-distance” introduced by Wu et al. (Math Comput Model 53:1331–1335, 2011a; IEICE Trans Inf Syst E94–D:226–232, 2011b), we propose new ranking and unranking algorithms for (k, m)-ary trees of order n in B-order using Zaks’ encoding. We show that both algorithms can be improved in $$\mathcal {O}(kmn)$$ time and $$\mathcal {O}(n)$$ space without building the precomputed table.

Approximation by genuine Gupta–Srivastava operators

In the present paper, we consider new operators, which is defined by Gupta and Srivastava (Eur J Pure Appl Math 11(3):575–579, 2018). They considered a general sequence of positive linear operators and gave the modified form of their previous operators (Neer et al. in Math Comput Model 37:1307–1315, 2003). As these operators preserve linear functions, we call these operators as genuine Gupta–Srivastava operators. Here we discuss some basic properties, direct results and rate of convergence of functions of bounded variation and weighted approximation.

Optimal control analysis of a tuberculosis model.

In this paper, we extend the model of Liu and Zhang (Math Comput Model 54:836-845, 2011) by incorporating three control terms and apply optimal control theory to the resulting model. Optimal control strategies are proposed to minimize both the disease burden and the intervention cost. We prove the existence and uniqueness of optimal control paths and obtain these optimal paths analytically using Pontryagin's Maximum Principle. We analyse our results numerically to compare various strategies of proposed controls. It is observed that implementation of three controls is most effective and less expensive among all the strategies. Thus, we conclude that in order to reduce tuberculosis threat all the three controls must be taken into consideration concurrently.

Stability for a New Discrete Ratio-Dependent Predator\u2013Prey System

The stability of a new two-species discrete ratio-dependent predator–prey system is considered. By using the linearization method, we obtain some sufficient conditions for the local stability of the positive equilibria. We also obtain a new sufficient condition to ensure that the positive equilibrium is globally asymptotically stable by using an iteration scheme and the comparison principle of difference equations, which generalizes what paper (Chen and Zhou in J Math Anal Appl 27:7358–7366, 2003) has done. The method given in this paper is new and very resultful comparing with articles (Damgaard in J Theor Biol 227:197–203, 2004; Edmunds in Theor Popul Biol 72:379–388, 2007; Fan and Wang in Math Comput Model 35:951–961, 2002; Muroya in J Math Anal Appl 330:24–33, 2007; Huo and Li in Appl Math Comput 153:337–351, 2004; Liao et al. in Appl Math Comput 190:500–509, 2007) and it can also be applied to study other global asymptotic stability for general multiple species discrete population systems. At the end of this paper, we present two open questions.

Two extended formulations for cardinality maximum flow network interdiction problem

We consider the maximum flow network interdiction problem in its cardinality case. There is an integer programming model for this problem by Wood (Math Comput Model 17 (1993), 1–18). Two types of valid inequalities have also been proposed (Altner et al., Oper Res Lett 38 (2010), 33–38 and Wood, Math Comput Model 17 (1993), 1–18) to strengthen the LP relaxation of the integer model. However, due to their combinatorial nature, the number of these inequalities are exponential. Here, we present an equivalent reformulation (extended formulation) for this problem which has a polynomial number of constraints. We also introduce new valid inequalities, and show that the corresponding reformulation of the LP relaxation of the integer model augmented with these inequalities, significantly decreases the integrality gap for a class of network interdiction problems with proven large integrality gaps. Numerical results for some benchmark as well as randomly generated instances are also reported. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 69(4), 367–377 2017

Demyelination patterns in a mathematical model of multiple sclerosis.

In this paper we derive a reaction-diffusion-chemotaxis model for the dynamics of multiple sclerosis. We focus on the early inflammatory phase of the disease characterized by activated local microglia, with the recruitment of a systemically activated immune response, and by oligodendrocyte apoptosis. The model consists of three equations describing the evolution of macrophages, cytokine and apoptotic oligodendrocytes. The main driving mechanism is the chemotactic motion of macrophages in response to a chemical gradient provided by the cytokines. Our model generalizes the system proposed by Calvez and Khonsari (Math Comput Model 47(7-8):726-742, 2008) and Khonsari and Calvez (PLos ONE 2(1):e150, 2007) to describe Baló's sclerosis, a rare and aggressive form of multiple sclerosis. We use a combination of analytical and numerical approaches to show the formation of different demyelinating patterns. In particular, a Turing instability analysis demonstrates the existence of a threshold value for the chemotactic coefficient above which stationary structures develop. In the case of subcritical transition to the patterned state, the numerical investigations performed on a 1-dimensional domain show the existence, far from the bifurcation, of complex spatio-temporal dynamics coexisting with the Turing pattern. On a 2-dimensional domain the proposed model supports the emergence of different demyelination patterns: localized areas of apoptotic oligodendrocytes, which closely fit existing MRI findings on the active MS lesion during acute relapses; concentric rings, typical of Baló's sclerosis; small clusters of activated microglia in absence of oligodendrocytes apoptosis, observed in the pathology of preactive lesions.

Finite difference numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh

Finding numerical solutions to the Troesch's problem is known to be challenging, especially when the sensitivity parameter $$\lambda $$? is large. In this manuscript, we propose a numerical method for solving the Troesch's problem which combines efficiency and accuracy, even for large sensitivity parameter. Our method can be summarized as a finite difference method formulated on a Shishkin mesh, which is piecewise uniform with smaller step size in the boundary layer in the neighborhood of the point $$x=1$$x=1. In fact, we treat the Troesch's problem as a singularly perturbed problem with a special transition point on the mesh. The transition point for the Shishkin mesh is first defined and computed. The application of the finite difference method on the piecewise uniform mesh leads to a nonlinear matrix system, which is solved numerically using the generalized Newton's method. While, existing numerical solvers suffer significantly when sensitivity parameter $$\lambda $$? becomes large and fail to provide accurate numerical solutions for $$\lambda >20$$?>20 (Chang, Appl Math Comput 216(11):3303---3306, 2010; Khuri and Sayfy, Math Comput Model 54(9):1907---1918, 2011; Raja, Inf Sci 279:860---873, 2014; Temimi, Appl Math Comput 219(2):521---529, 2012), our method provides accurate numerical solution for large values of this sensitivity parameter, up to $$\lambda =100$$?=100. Moreover, the implementation of the method is straight forward and the computation cost is low. Numerical experiments show that the method provides accurate solutions for different values of the sensitivity parameter $$\lambda $$?.

Commentaries on Viewpoint: A call for research to assess and promote functional resilience in astronaut crews

Commentaries on Viewpoint: A call for research to assess and promote functional resilience in astronaut crews

An efficient numerical algorithm based on Haar wavelet for solving a class of linear and nonlinear nonlocal boundary-value problems

Based on Haar wavelet a new numerical method for the solution of a class of nth-order boundary-value problems (BVPs) with nonlocal boundary conditions is proposed. This new method is an extension of the Haar wavelet method (Siraj-ul-Islam et al. in Math Comput Model 50:1577---1590, 2010; Int J Therm Sci 50:686---697, 2011) from BVPs with local boundary conditions to BVPs with a class of nonlocal boundary conditions. A major advantage of the proposed method is that it is applicable to both linear and nonlinear BVPs. The method is tested on BVPs of third- and fourth-order. The numerical results are compared with an existing method in the literature and analytical solutions. The numerical experiments demonstrate the accuracy and efficiency of the proposed method.

On average values of convex functions

The aim of this paper is to give an extension of an inequality proved by Wulbert (Math Comput Model 37:1383–1391, 2003, Lemma 2.5) and to define Stolarsky type means as an application of this inequality. Further, we discuss some properties of averages of a continuous convex function, some consequences of a double inequality given by Wulbert (Math Comput Model 37:1383–1391, 2003, Theorem 3.3) and obtain improvement results of Wulbert (Math Comput Model 37:1383–1391, 2003, Corollary 4.3).

A mathematical analysis of physiological and morphological aspects of wound closure

A computational algorithm to study the evolution of complex wound morphologies is developed based on a model of wound closure by cell mitosis and migration due to Adam [Math Comput Model 30(5-6):23-32, 1999]. A detailed analysis of the model provides estimated values for the incubation and healing times. Furthermore, a set of inequalities are defined which demarcate conditions of complete, partial and non-healing. Numerical results show a significant delay in the healing progress whenever diffusion of the epidermic growth factor responsible for cell mitosis is slower than cell migration. Results for general wound morphologies show that healing is always initiated at regions with high curvatures and that the evolution of the wound is very sensitive to physiological parameters.

Tumor evasion from immune control: Strategies of a MISS to become a MASS

We biologically describe the phenomenon of the evasion of tumors from immune surveillance where tumor cells, initially constrained to exist in a microscopic steady state (MISS) elaborate strategies to evade from the immune control and to reach a macroscopic steady state (MASS). We, then, describe “evasion” as a long term loss of equilibrium in a framework of prey–predator-like models with adiabatic varying parameters, whose changes reflect the evolutionary adaptation of the tumor in a “hostile” environment by means of the elaboration of new strategies of survival. Similarities and differences between the present work and the interesting seminal paper [Kuznetsov VA, Knott GD. Modeling tumor regrowth and immunotherapy. Math Comput Model 2001;33:1275–87] are discussed. We also propose and study a model of clonal resistance to the immune control with slowly varying adaptive mutation parameter.