#### American Journal of Computational Mathematics

Journal Information

ISSN / EISSN :
21611203 / 21611211

Current Publisher: Scientific Research Publishing, Inc. (10.4236)

Total articles ≅ 373

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#### Latest articles in this journal

Published: 7 January 2020

American Journal of Computational Mathematics, Volume 10, pp 118-146; doi:10.4236/ajcm.2020.101008

**Abstract:**

With the widespread application of distributed systems, many problems need to be solved urgently. How to design distributed optimization strategies has become a research hotspot. This article focuses on the solution rate of the distributed convex optimization algorithm. Each agent in the network has its own convex cost function. We consider a gradient-based distributed method and use a push-pull gradient algorithm to minimize the total cost function. Inspired by the current multi-agent consensus cooperation protocol for distributed convex optimization algorithm, a distributed convex optimization algorithm with finite time convergence is proposed and studied. In the end, based on a fixed undirected distributed network topology, a fast convergent distributed cooperative learning method based on a linear parameterized neural network is proposed, which is different from the existing distributed convex optimization algorithms that can achieve exponential convergence. The algorithm can achieve finite-time convergence. The convergence of the algorithm can be guaranteed by the Lyapunov method. The corresponding simulation examples also show the effectiveness of the algorithm intuitively. Compared with other algorithms, this algorithm is competitive.

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 167-220; doi:10.4236/ajcm.2020.102010

**Abstract:**

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 275-289; doi:10.4236/ajcm.2020.102015

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 252-265; doi:10.4236/ajcm.2020.102013

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 90-99; doi:10.4236/ajcm.2020.101006

**Abstract:**

Finite Element Method (FEM), based on p and h versions approach, and the Adomians decomposition algorithm (ADM) are introduced for solving the Emden-Fowler Equation. A number of special cases of p and h versions of FEM are introduced. Several iterated forms of the ADM are considered also. To demonstrate the efficiency of both methods, the numerical solutions of different examples are compared for both methods with the analytical solutions. It is observed that the results obtained by FEM are quite satisfactory and more accurate than ADM. Moreover, the FEM method is applicable for a wide range of classes including the singularity cases with the given special treatments by the FEM. Comparing the results with the existing true solutions shows that the FEM approach is highly accurate and converges rapidly.

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 1-22; doi:10.4236/ajcm.2020.101001

**Abstract:**

In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles.

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 31-42; doi:10.4236/ajcm.2020.101003

**Abstract:**

In this paper, the global stability of free smoking equilibrium point was evaluated and presented graphically. The linear stability of a developed mathematical model illustrates the effect on the population of chain, mild and passive smokers. MATLAB programming was used to simulate the solutions, the reproduction number R0 and the nature of the equilibria.

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 410-424; doi:10.4236/ajcm.2020.103022

**Abstract:**

One of the most important activities in data science is defining a membership function in fuzzy system. Although there are few ways to describe membership function like artificial neural networks, genetic algorithms etc.; they are very complex and time consuming. On the other hand, the presence of outlier in a data set produces deceptive results in the modeling. So it is important to detect and eliminate them to prevent their negative effect on the modeling. This paper describes a new and simple way of constructing fuzzy membership function by using five-number summary of a data set. Five states membership function can be created in this new method. At the same time, if there is any outlier in the data set, it can be detected with the help of this method. Usually box plot is used to identify the outliers of a data set. So along with the new approach, the box plot has also been drawn so that it is understood that the results obtained in the new method are accurate. Several real life examples and their analysis have been discussed with graph to demonstrate the potential of the proposed method. The results obtained show that the proposed method has given good results. In the case of outlier, the proposed method and the box plot method have shown similar results. Primary advantage of this new procedure is that it is not as expensive as neural networks, and genetic algorithms.

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 355-381; doi:10.4236/ajcm.2020.103019

Published: 1 January 2020

American Journal of Computational Mathematics, Volume 10, pp 147-166; doi:10.4236/ajcm.2020.101009

**Abstract:**

In this paper, we study a drive-response discrete-time dynamical system which has been coupled using convex functions and we introduce a synchronization threshold which is crucial for the synchronizing procedure. We provide one application of this type of coupling in synchronized cycles of a generalized Nicholson-Bailey model. This model demonstrates a rich cascade of complex dynamics from stable fixed point to periodic orbits, quasi periodic orbits and chaos. We explain how this way of coupling makes these two chaotic systems starting from very different initial conditions, quickly get synchronized. We investigate the qualitative behavior of GNB model and its synchronized model using time series analysis and its long time dynamics by the help of bifurcation diagram.