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Numerical Algebra, Control & Optimization

December 2019 , Volume 9 , Issue 4

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Numerical solution with analysis of HIV/AIDS dynamics model with effect of fusion and cure rate
Praveen Kumar Gupta and Ajoy Dutta
2019, 9(4): 393-399 doi: 10.3934/naco.2019038 +[Abstract](278) +[HTML](124) +[PDF](311.17KB)
Abstract:

The main objective of this manuscript is to study the dynamical behaviour and numerical solution of a HIV/AIDS dynamics model with fusion effect and cure rate. Local and global asymptotic stability of the model is established by Routh-Hurwitz criterion and Lyapunov functional method for infection-free equilibrium point. The numerical solutions of the model has also examined for support of analysis, through Mathematica software.

Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation
Fatemeh Ahangari
2019, 9(4): 401-412 doi: 10.3934/naco.2019039 +[Abstract](384) +[HTML](191) +[PDF](374.08KB)
Abstract:

Concerning holonomy theory or in the context of the existence of parallel spinors, Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation have considerable significance. In this paper, we have comprehensively concentrated on conformal deformations of a particular class of four dimensional Lorentzian manifolds with indecomposable, non-irreducible holonomy representation which admit a recurrent light-like vector field. This type of Lorentzian manifolds are denoted by pr-waves and their holonomy algebra is contained in the parabolic algebra \begin{document}$ \big(\mathbb{R}\oplus \mbox{so(2)}\big)\ltimes \mathbb{R}^2 $\end{document}. Moreover, it is mainly illustrated that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.

Onset of Benard-Marangoni instabilities in a double diffusive binary fluid layer with temperature-dependent viscosity
Nurul Hafizah Zainal Abidin, Nor Fadzillah Mohd Mokhtar and Zanariah Abdul Majid
2019, 9(4): 413-421 doi: 10.3934/naco.2019040 +[Abstract](222) +[HTML](147) +[PDF](529.57KB)
Abstract:

The effect of temperature-dependent viscosity in a horizontal double diffusive binary fluid layer is investigated. When the layer is heated from below, the convection of Benard-Marangoni will start to exists. Linear stability analysis is performed and the eigenvalues from few cases of boundary conditions were obtained. Galerkin method were used to solve the numerical calculation and marginal stability curve is obtained. Results shows that an increase of temperature-dependent viscosity will destabilized the system. The impact of double diffusive coefficients are also revealed. It is found that the effect of Soret parameter exhibits destabilizing reaction on the system while an opposite response is noted with an increase of Dufour parameter.

Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model
Najwa Najib, Norfifah Bachok, Norihan Md Arifin and Fadzilah Md Ali
2019, 9(4): 423-431 doi: 10.3934/naco.2019041 +[Abstract](296) +[HTML](164) +[PDF](635.86KB)
Abstract:

The study on stagnation boundary layer flow in nanofluid over stretching/shrinking sheet with the effect of slip at the boundary was considered by applying the Buongiorno's model. The partial differential equations of the governing equations were transformed into ordinary differential equations by using appropriate similarity transformation in order to obtain the similarity equations. The equations then were substituted into bvp4c code in Matlab software to get the numerical results. The results of skin friction coefficient, heat transfer coefficient as well as mass transfer coefficient on the governing parameters such as slip parameter, Brownian motion parameter, and thermophoresis parameter are shown graphically. The presence of slip parameter is significantly affected the skin friction, heat and mass transfer coefficient. The smallest number of Brownian motion is sufficient to increase the heat transfer coefficient while largest number of thermophoresis parameter is required to increase mass transfer coefficient. The stability analysis results expressed that the first solution is stable and physically realizable whereas the second solution is not.

Numerical solutions of Volterra integro-differential equations using General Linear Method
Faranak Rabiei, Fatin Abd Hamid, Zanariah Abd Majid and Fudziah Ismail
2019, 9(4): 433-444 doi: 10.3934/naco.2019042 +[Abstract](284) +[HTML](166) +[PDF](503.4KB)
Abstract:

In this paper, a third order General Linear Method for finding the numerical solution of Volterra integro-differential equation is considered. The order conditions of the proposed method are derived based on techniques of B-series and 'rooted trees'. The integral operator in Volterra integro-differential equation approximated using Simpson's rule and Lagrange interpolation is discussed. To illustrate the efficiency of third order General Linear Method, we compare the method with a third order Runge-Kutta method.

System of generalized mixed nonlinear ordered variational inclusions
Salahuddin
2019, 9(4): 445-460 doi: 10.3934/naco.2019026 +[Abstract](514) +[HTML](273) +[PDF](403.11KB)
Abstract:

In this paper, we consider a system of generalized mixed nonlinear ordered variational inclusions in partially ordered Banach spaces and suggest an algorithm for a solution of the considered system. We prove an existence and convergence result for the solution of the system of generalized mixed nonlinear ordered variational inclusions.

$ \theta $ scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations
Yang Wang and Yi-fu Feng
2019, 9(4): 461-481 doi: 10.3934/naco.2019027 +[Abstract](501) +[HTML](264) +[PDF](417.42KB)
Abstract:

In this article, a \begin{document}$ \theta $\end{document} scheme based on wavelet-like incremental unknowns (WIU) is presented for a class of porous medium diffusion-type equations. Through some important norm inequalities, we prove the stability of \begin{document}$ \theta $\end{document} scheme. Compared to the classical scheme, the stability conditions are improved. Numerical results show that the \begin{document}$ \theta $\end{document} scheme based on the WIU decomposition is efficient.

A preconditioned SSOR iteration method for solving complex symmetric system of linear equations
Tahereh Salimi Siahkolaei and Davod Khojasteh Salkuyeh
2019, 9(4): 483-492 doi: 10.3934/naco.2019033 +[Abstract](523) +[HTML](301) +[PDF](408.9KB)
Abstract:

We present a preconditioned version of the symmetric successive overrelaxation (SSOR) iteration method for a class of complex symmetric linear systems. The convergence results of the proposed method are established and conditions under which the spectral radius of the iteration matrix of the method is smaller than that of the SSOR method are analyzed. Numerical experiments illustrate the theoretical results and depict the efficiency of the new iteration method.

A hybrid parametrization approach for a class of nonlinear optimal control problems
M. Alipour, M. A. Vali and A. H. Borzabadi
2019, 9(4): 493-506 doi: 10.3934/naco.2019037 +[Abstract](698) +[HTML](317) +[PDF](395.4KB)
Abstract:

In this paper, a suitable hybrid iterative scheme for solving a class of non-linear optimal control problems (NOCPs) is proposed. The technique is based upon homotopy analysis and parametrization methods. Actually an appropriate parametrization of control is applied and state variables are computed using homotopy analysis method (HAM). Then performance index is transformed by replacing new control and state variables. The results obtained from the given method are compared with the results which are obtained using the spectral homotopy analysis method (SHAM), homotopy perturbation method (HPM), optimal homotopy perturbation method (OHPM), modified variational iteration method (MVIM) and differential transformations. The existence and uniqueness of the solution are presented. The comparison and ability of the given approach is illustrated via two examples.

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