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Discrete and Continuous Dynamical Systems - S

July 2021 , Volume 14 , Issue 7

Issue on modelling and applied mathematics in nature

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Abdon Atangana and Zakia Hammouch
2021, 14(7): i-i doi: 10.3934/dcdss.2021073 +[Abstract](677) +[HTML](110) +[PDF](67.41KB)
Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme
Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari and Dumitru Baleanu
2021, 14(7): 2025-2039 doi: 10.3934/dcdss.2020402 +[Abstract](2065) +[HTML](465) +[PDF](786.93KB)

This paper develops a numerical scheme for finding the approximate solution of space fractional order of the diffusion equation (SFODE). Firstly, the compact finite difference (CFD) with convergence order \begin{document}$ \mathcal{O}(\delta \tau ^{2}) $\end{document} is used for discretizing time derivative. Afterwards, the spatial fractional derivative is approximated by the Chebyshev collocation method of the fourth kind. Furthermore, time-discrete stability and convergence analysis are presented. Finally, two examples are numerically investigated by the proposed method. The examples illustrate the performance and accuracy of our method compared to existing methods presented in the literature.

A new application of the reproducing kernel method
Ali Akgül
2021, 14(7): 2041-2053 doi: 10.3934/dcdss.2020261 +[Abstract](1177) +[HTML](322) +[PDF](314.66KB)

We give a new implementation of the reproducing kernel method to investigate difference equations in this paper. We obtain the solutions in terms of convergent series. The method of obtaining the approximate solution in form of an algorithm is presented. We demonstrate some experiments to prove the accuracy of the technique.

Fractional and fractal advection-dispersion model
Amy Allwright, Abdon Atangana and Toufik Mekkaoui
2021, 14(7): 2055-2074 doi: 10.3934/dcdss.2021061 +[Abstract](700) +[HTML](154) +[PDF](1320.81KB)

A fractal advection-dispersion equation and a fractional space-time advection-dispersion equation have been developed to improve the simulation of groundwater transport in fractured aquifers. The space-time fractional advection-dispersion simulation is limited due to complex algorithms and the computational power required; conversely, the fractal advection-dispersion equation can be solved simply, yet only considers the fractal derivative in space. These limitations lead to combining these methods, creating a fractional and fractal advection-dispersion equation to provide an efficient non-local, in both space and time, modeling tool. The fractional and fractal model has two parameters, fractional order (\begin{document}$ \alpha $\end{document}) and fractal dimension (\begin{document}$ \beta $\end{document}), where simulations are valid for specific combinations. The range of valid combinations reduces with decreasing fractional order and fractal dimension, and a final recommendation of \begin{document}$ \; 0.7 \leq \alpha, \beta \leq 1 $\end{document} is made. The fractional and fractal model provides a flexible tool to model anomalous diffusion, where the fractional order controls the breakthrough curve peak, and the fractal dimension controls the position of the peak and tailing effect. These two controls potentially provide tools to improve the representation of anomalous breakthrough curves that cannot be described by the classical model.

Parallelization of a finite volumes discretization for anisotropic diffusion problems using an improved Schur complement technique
Hassan Belhadj, Samir Khallouq and Mohamed Rhoudaf
2021, 14(7): 2075-2099 doi: 10.3934/dcdss.2020260 +[Abstract](1401) +[HTML](405) +[PDF](1681.67KB)

We present in this paper a new algorithm combining a finite volume method with an improved Schur complement technique to solve \begin{document}$ 2D $\end{document} anisotropic diffusion problems on general meshes. After having proved the convergence of the finite volume method, we have given a description of the proposed algorithm in the case of two nonoverlapping subdomains. Several numerical tests are achieved which illustrate the theoretical results of convergence of the finite volume method and show the advantages of the proposed algorithm.

An analysis of tuberculosis model with exponential decay law operator
Ebenezer Bonyah and Fatmawati
2021, 14(7): 2101-2117 doi: 10.3934/dcdss.2021057 +[Abstract](744) +[HTML](102) +[PDF](734.98KB)

In this paper, we explore the dynamics of tuberculosis (TB) epidemic model that includes the recruitment rate in both susceptible and infected population. Stability and sensitivity analysis of the classical TB model is carried out. Caputo-Fabrizio (CF) operator is then used to explain the dynamics of the TB model. The concept of fixed point theory is employed to obtain the existence and uniqueness of the solution of the TB model in the light of CF operator. Numerical simulations based on Homotopy Analysis Transform Method (HATM) and padé approximations are performed to obtain qualitative information on the model. Numerical solutions depict that the order of the fractional derivative has great dynamics of the TB model.

More new results on integral inequalities for generalized $ \mathcal{K} $-fractional conformable Integral operators
Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor and Humaira Kalsoom
2021, 14(7): 2119-2135 doi: 10.3934/dcdss.2021063 +[Abstract](958) +[HTML](103) +[PDF](410.78KB)

This paper aims to investigate the several generalizations by newly proposed generalized \begin{document}$ \mathcal{K} $\end{document}-fractional conformable integral operator. Based on these novel ideas, we derived a novel framework to study for \begin{document}$ \breve{C} $\end{document}eby\begin{document}$ \breve{s} $\end{document}ev and P\begin{document}$ \acute{o} $\end{document}lya-Szeg\begin{document}$ \ddot{o} $\end{document} type inequalities by generalized \begin{document}$ \mathcal{K} $\end{document}-fractional conformable integral operator. Several special cases are apprehended in the light of generalized fractional conformable integral. This novel strategy captures several existing results in the relative literature. We also aim at showing important connections of the results here with those including Riemann-Liouville fractional integral operator.

Bounded perturbation for evolution equations with a parameter & application to population dynamics
Emile Franc Doungmo Goufo
2021, 14(7): 2137-2150 doi: 10.3934/dcdss.2020177 +[Abstract](614) +[HTML](104) +[PDF](625.88KB)

Evolution equations using derivatives of fractional order like Caputo's derivative or Riemann-Liouville's derivative have been intensively analyzed in numerous works. But the classical bounded perturbation theorem has been proven not to be in general true for these models, especially for solution operators of evolution equations with fractional order derivative \begin{document}$ \alpha $\end{document} less than \begin{document}$ 1 $\end{document} (\begin{document}$ 0<\alpha<1 $\end{document}), as shown by the example in the next section. This paper proposes an alternative way of dealing with this issue. We make use of the conventional time derivative with a new parameter to show the perturbations by bounded linear operators for linear evolution equations when the derivative order is less than one. The new parameter which happens to be fractional, characterizes the so-called \begin{document}$ \beta $\end{document}-derivative. Its fractional order parameter allows the use of concepts like revamped time to provide a relation between both strongly continuous two-parameter solution operators involved in the perturbation process. To validate the theory, we use an application to population dynamics and perform some numerical simulations that reveal some consistency with the expected results.

Mathematical model of diabetes and its complication involving fractional operator without singular kernal
Ravi Shanker Dubey and Pranay Goswami
2021, 14(7): 2151-2161 doi: 10.3934/dcdss.2020144 +[Abstract](2702) +[HTML](928) +[PDF](575.88KB)

Diabetes is one of the burning issues of the whole world. It effected the world population rapidly. According to the WHO approx 415 million people are living with diabetes in the world and this figure is expected to rise up to 642 million by 2040. World various organizations raise their voice against the dire facts about the increasing graph of diabetes and its complicated patients. In this paper authors define the fractional model of diabetes and its complications involving to fractional operator with exponential kernel. The authors discuss the existence of the solution by using fixed point theorem and also show the uniqueness of the solution. To validate the model's efficiency the authors provided numerical simulation by using HPM. To strengthen the model the results have been presented in the form of graphs.

Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion
Anouar El Harrak, Amal Bergam, Tri Nguyen-Huu, Pierre Auger and Rachid Mchich
2021, 14(7): 2163-2181 doi: 10.3934/dcdss.2021055 +[Abstract](703) +[HTML](210) +[PDF](478.15KB)

The main goal of this paper is to adapt a class of complexity reduction methods called aggregation of variables methods to the construction of reduced models of two-time reaction-diffusion-chemotaxis models of spatially structured populations and to provide an error bound of the approximate dynamics. Aggregation of variables methods are general techniques that allow reducing the dimension of a mathematical dynamical system. Here we reduce a system of Partial Differential Equations to a simpler Ordinary Differential Equation system, provided that the evolution processes occur at two different time scales: a slow one for the demography and a fast one for migrations and chemotaxis, with a ratio \begin{document}$ \varepsilon>0 $\end{document} small enough. We give an approximation of the error between solutions of both original and reduced model for a generic function representing the demography. Finally, we provide an optimization of the error bound and validate numerically this result for a spatial inter-specific model with constant diffusion and population growth given by a logistic law in population dynamics.

A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics
Anouar El Harrak, Hatim Tayeq and Amal Bergam
2021, 14(7): 2183-2197 doi: 10.3934/dcdss.2021062 +[Abstract](916) +[HTML](226) +[PDF](438.14KB)

This work gives a posteriori error estimates for a finite volume implicit scheme, applied to a two-time nonlinear reaction-diffusion problem in population dynamics, whose evolution processes occur at two different time scales, represented by a parameter \begin{document}$ \varepsilon>0 $\end{document} small enough. This work consists of building error indicators concerning time and space approximations and using them as a tool of adaptive mesh refinement in order to find approximate solutions to such models, in population dynamics, that are often hard to be handled analytically and also to be approximated numerically using the classical approach.

An application of the theoretical results is provided to emphasize the efficiency of our approach compared to the classical one for a spatial inter-specific model with constant diffusivity and population growth given by a logistic law in population dynamics.

System response of an alcoholism model under the effect of immigration via non-singular kernel derivative
Fırat Evirgen, Sümeyra Uçar, Necati Özdemir and Zakia Hammouch
2021, 14(7): 2199-2212 doi: 10.3934/dcdss.2020145 +[Abstract](645) +[HTML](107) +[PDF](624.86KB)

In this study, we aim to comprehensively investigate a drinking model connected to immigration in terms of Atangana-Baleanu derivative in Caputo type. To do this, we firstly extend the model describing drinking model by changing the derivative with time fractional derivative having Mittag-Leffler kernel. The existence and uniqueness of the drinking model solutions together with the stability analysis is shown by the help of Banach fixed point theorem. The special solution of the model is investigated using the Sumudu transformation and then, we present some numerical simulations for the different fractional orders to emphasize the effectiveness of the used derivative.

Melnikov analysis of the nonlocal nanobeam resting on fractional-order softening nonlinear viscoelastic foundations
Guy Joseph Eyebe, Betchewe Gambo, Alidou Mohamadou and Timoleon Crepin Kofane
2021, 14(7): 2213-2228 doi: 10.3934/dcdss.2020252 +[Abstract](1807) +[HTML](584) +[PDF](751.31KB)

In the present study, the dynamics of nanobeam resting on fractional order softening nonlinear viscoelastic pasternack foundations is studied. The Hamilton principle is used to derive the nonlinear equation of the motion. Approximate analytical solution is obtained by applying the standard averaging method. The Melnikov method is used to investigate the chaotic behaviors of device, the critical curve separating the chaotic and non-chaotic regions are found. It is shown that the distance between chaotic region and non-chaotic region in this kind of structure depends strongly on the fractional order parameter.

Stochastic dynamics of the FitzHugh-Nagumo neuron model through a modified Van der Pol equation with fractional-order term and Gaussian white noise excitation
Boris Anicet Guimfack, Conrad Bertrand Tabi, Alidou Mohamadou and Timoléon Crépin Kofané
2021, 14(7): 2229-2243 doi: 10.3934/dcdss.2020397 +[Abstract](1701) +[HTML](476) +[PDF](1877.12KB)

The stochastic response of the FitzHugh-Nagumo model is addressed using a modified Van der Pol (VDP) equation with fractional-order derivative and Gaussian white noise excitation. Via the generalized harmonic balance method, the term related to fractional derivative is splitted into the equivalent quasi-linear dissipative force and quasi-linear restoring force, leading to an equivalent VDP equation without fractional derivative. The analytical solutions for the equivalent stochastic equation are then investigated through the stochastic averaging method. This is thereafter compared to numerical solutions, where the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity are used to characterized the dynamical behaviors of the system. A satisfactory agreement is found between the two approaches, which confirms the accuracy of the used analytical method. It is also found that changing the fractional-order parameter and the intensity of the Gaussian white noise induces P-bifurcation.

Dynamical behaviors and oblique resonant nonlinear waves with dual-power law nonlinearity and conformable temporal evolution
Md. Golam Hafez, Sayed Allamah Iqbal, Asaduzzaman and Zakia Hammouch
2021, 14(7): 2245-2260 doi: 10.3934/dcdss.2021058 +[Abstract](1087) +[HTML](97) +[PDF](1535.53KB)

In this article, the oblique resonant traveling waves and dynamical behaviors of (2+1)-dimensional Nonlinear Schrödinger equation along with dual-power law nonlinearity, and fractal conformable temporal evolution are reported. The considered equation is converted to an ordinary differential equation by taking the traveling variable wave transform and properties of Khalil's conformable derivative into account. The modified Kudryashov method is implemented to divulge the oblique resonant traveling wave of such an equation. It is found that the obliqueness is only affected on width, but not on amplitude and phase patriots of resonant nonlinear propagating wave dynamics. The research outcomes are very helpful for analyzing the obliquely propagating nonlinear resonant wave phenomena and their dynamical behaviors in several nonlinear systems having Madelung fluids and optical bullets.

A novel model for the contamination of a system of three artificial lakes
Veysel Fuat Hatipoğlu
2021, 14(7): 2261-2272 doi: 10.3934/dcdss.2020176 +[Abstract](2020) +[HTML](782) +[PDF](3363.64KB)

In this study, a new model has been developed to monitor the contamination in connected three lakes. The model has been motivated by two biological models, i.e. cell compartment model and lake pollution model. Haar wavelet collocation method has been proposed for the numerical solutions of the model containing a system of three linear differential equations. In addition to the solutions of the system, convergence analysis has been briefly given for the proposed method. The contamination in each lake has been investigated by considering three different pollutant input cases, namely impulse imposed pollutant source, exponentially decaying imposed pollutant source, and periodic imposed pollutant source. Each case has been illustrated with a numerical example and results are compared with the exact ones. Regarding the results in each case it has been seen that, Haar wavelet collocation method is an efficient algorithm to monitor the contamination of a system of lakes problem.

A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation
Masoumeh Hosseininia, Mohammad Hossein Heydari and Carlo Cattani
2021, 14(7): 2273-2295 doi: 10.3934/dcdss.2020295 +[Abstract](1834) +[HTML](420) +[PDF](5372.34KB)

In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions for nonlinear variable-order time fractional two-dimensional (2D) Schrödinger equation. First, the variable-order time fractional derivative involved in the considered problem is approximated via the finite difference technique. Then, by help of the finite difference scheme and the theta-weighted method, a recursive algorithm is derived for the problem under examination. After that, the real functions available in the real and imaginary parts of the unknown solution of the problem are expanded via the 2D LWs. Finally, by applying the operational matrices of derivative, the solution of the problem is transformed to the solution of a linear system of algebraic equations in each time step which can simply be solved. In the proposed method, acceptable approximate solutions are achieved by employing only a small number of the basis functions. To illustrate the applicability, validity and accuracy of the wavelet method, some numerical test examples are solved using the suggested method. The achieved numerical results reveal that the method established based on the 2D LWs is very easy to implement, appropriate and accurate in solving the proposed model.

New class of volterra integro-differential equations with fractal-fractional operators: Existence, uniqueness and numerical scheme
Seda İğret Araz
2021, 14(7): 2297-2309 doi: 10.3934/dcdss.2021053 +[Abstract](883) +[HTML](116) +[PDF](353.52KB)

In this paper, we introduce a new fractional integro-differential equation involving newly introduced differential and integral operators so-called fractal-fractional derivatives and integrals. We present a numerical scheme that is convenient for obtaining solution of such equations. We give the general conditions for the existence and uniqueness of the solution of the considered equation using Banach fixed-point theorem. Both the suggested new equation and new numerical scheme will considerably contribute for our readers in theory and applications.

Abundant novel solutions of the conformable Lakshmanan-Porsezian-Daniel model
Hajar Farhan Ismael, Haci Mehmet Baskonus and Hasan Bulut
2021, 14(7): 2311-2333 doi: 10.3934/dcdss.2020398 +[Abstract](2124) +[HTML](688) +[PDF](3146.08KB)

In this paper, three images of nonlinearity to the fractional Lakshmanan Porsezian Daniel model in birefringent fibers are investigated. The new bright, periodic wave and singular optical soliton solutions are constructed via the \begin{document}$ \left( m+\frac{G'}{G} \right) $\end{document} expansion method, which are applicable to the dynamics within the optical fibers. All solutions are novel compared with solutions obtained via different methods. All solutions verify the conformable Lakshmanan-Porsezian-Daniel model and also, for the existence the constraint conditions are utilized. Moreover, 2D and 3D for all solutions are plotted to more understand its physical characteristics.

Lyapunov type inequality in the frame of generalized Caputo derivatives
Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad, Saed F. Mallak and Hussam Alrabaiah
2021, 14(7): 2335-2355 doi: 10.3934/dcdss.2020212 +[Abstract](867) +[HTML](122) +[PDF](441.92KB)

In this paper, we establish the Lyapunov-type inequality for boundary value problems involving generalized Caputo fractional derivatives that unite the Caputo and Caputo-Hadamrad fractional derivatives. An application about the zeros of generalized types of Mittag-Leffler functions is given.

Electromagnetic waves described by a fractional derivative of variable and constant order with non singular kernel
Krunal B. Kachhia and Abdon Atangana
2021, 14(7): 2357-2371 doi: 10.3934/dcdss.2020172 +[Abstract](1193) +[HTML](342) +[PDF](314.67KB)

The concept of differential operator with variable order has attracted attention of many scholars due to their abilities to capture more complexities like anomalous diffusion. While these differential operators are useful in real life, they can only be handled numerically. In this work, we used a newly introduced variable order differential operators that can be used analytically and numerically, has connection with all integral transform to model some interesting mathematical models arising in electromagnetic wave in plasma and dielectric. The differential operators used are non-singular and have the crossover properties therefore the models studied can explain the propagation of the wave in two different layers which cannot be achieved with those differential variable order operators with singular kernels. Using the Laplace transform and its connection with the new differential operator, we derive the exact solution of the models under investigation.

Numerical treatment of Gray-Scott model with operator splitting method
Berat Karaagac
2021, 14(7): 2373-2386 doi: 10.3934/dcdss.2020143 +[Abstract](1886) +[HTML](372) +[PDF](616.61KB)

This article focuses on the numerical solution of a classical, irreversible Gray Scott reaction-diffusion system describing the kinetics of a simple autocatalytic reaction in an unstirred ow reactor. A novel finite element numerical scheme based on B-spline collocation method is developed to solve this model. Before applying finite element method, "strang splitting" idea especially popularized for reaction-diffusion PDEs has been applied to the model. Then, using the underlying idea behind finite element approximation, the domain of integration is partitioned into subintervals which is sought as the basis for the B-spline approximate solution. Thus, the partial derivatives are transformed into a system of algebraic equations. Applicability and accuracy of this method is justified via comparison with the exact solution and calculating both the error norms \begin{document}$ L_2 $\end{document} and \begin{document}$ L_\infty $\end{document}. Numerical results arising from the simulation experiments are also presented.

Exact analytical solutions of fractional order telegraph equations via triple Laplace transform
Rahmat Ali Khan, Yongjin Li and Fahd Jarad
2021, 14(7): 2387-2397 doi: 10.3934/dcdss.2020427 +[Abstract](1286) +[HTML](361) +[PDF](288.47KB)

In this paper, we study initial/boundary value problems for \begin{document}$ 1+1 $\end{document} dimensional and \begin{document}$ 1+2 $\end{document} dimensional fractional order telegraph equations. We develop the technique of double and triple Laplace transforms and obtain exact analytical solutions of these problems. The techniques we develop are new and are not limited to only telegraph equations but can be used for exact solutions of large class of linear fractional order partial differential equations

Interpolation of exponential-type functions on a uniform grid by shifts of a basis function
Jeremy Levesley, Xinping Sun, Fahd Jarad and Alexander Kushpel
2021, 14(7): 2399-2416 doi: 10.3934/dcdss.2020403 +[Abstract](1435) +[HTML](429) +[PDF](334.73KB)

In this paper, we present a new approach to solving the problem of interpolating a continuous function at \begin{document}$ (n+1) $\end{document} equally-spaced points in the interval \begin{document}$ [0, 1] $\end{document}, using shifts of a kernel on the \begin{document}$ (1/n) $\end{document}-spaced infinite grid. The archetypal example here is approximation using shifts of a Gaussian kernel. We present new results concerning interpolation of functions of exponential type, in particular, polynomials on the integer grid as a step en route to solve the general interpolation problem. For the Gaussian kernel we introduce a new class of polynomials, closely related to the probabilistic Hermite polynomials and show that evaluations of the polynomials at the integer points provide the coefficients of the interpolants. Finally we give a closed formula for the Gaussian interpolant of a continuous function on a uniform grid in the unit interval (assuming knowledge of the discrete moments of the Gaussian).

A Sturm-Liouville approach for continuous and discrete Mittag-Leffler kernel fractional operators
Raziye Mert, Thabet Abdeljawad and Allan Peterson
2021, 14(7): 2417-2434 doi: 10.3934/dcdss.2020171 +[Abstract](1977) +[HTML](767) +[PDF](410.0KB)

In this work, we use integration by parts formulas derived for fractional operators with Mittag-Leffler kernels to formulate and investigate fractional Sturm-Liouville Problems (\begin{document}$ FSLPs $\end{document}). We analyze the self-adjointness, eigenvalue and eigenfunction properties of the associated Fractional Sturm-Liouville Operators (\begin{document}$ FSLOs $\end{document}). The discrete analogue of the obtained results is formulated and analyzed by following nabla analysis.

A fuzzy inventory model for Weibull deteriorating items under completely backlogged shortages
Deepak Kumar Nayak, Sudhansu Sekhar Routray, Susanta Kumar Paikray and Hemen Dutta
2021, 14(7): 2435-2453 doi: 10.3934/dcdss.2020401 +[Abstract](1577) +[HTML](504) +[PDF](433.83KB)

In this paper, a fuzzy stock replenishment policy implemented for inventory items that follows linear demand and Weibull deterioration under completely backlogged shortages. Moreover, to minimize the aggregate expense per unit time, the fuzzy optimal solution is obtained using general mathematical techniques by considering hexagonal fuzzy numbers and graded mean preference integration strategy. Finally, the complete exposition of the model is provided by numerical examples and sensitivity behavior of the associated parameters.

Fractional Adams-Bashforth scheme with the Liouville-Caputo derivative and application to chaotic systems
Kolade M. Owolabi, Abdon Atangana and Jose Francisco Gómez-Aguilar
2021, 14(7): 2455-2469 doi: 10.3934/dcdss.2021060 +[Abstract](1028) +[HTML](118) +[PDF](13744.11KB)

A recently proposed numerical scheme for solving nonlinear ordinary differential equations with integer and non-integer Liouville-Caputo derivative is applied to three systems with chaotic solutions. The Adams-Bashforth scheme involving Lagrange interpolation and the fundamental theorem of fractional calculus. We provide the existence and uniqueness of solutions, also the convergence result is stated. The proposed method is applied to several examples that are shown to have unique solutions. The scheme converges to the classical Adams-Bashforth method when the fractional orders of the derivatives converge to integers.

Finite element method for two-dimensional linear advection equations based on spline method
Kai Qu, Qi Dong, Chanjie Li and Feiyu Zhang
2021, 14(7): 2471-2485 doi: 10.3934/dcdss.2021056 +[Abstract](807) +[HTML](107) +[PDF](1642.67KB)

A new method for some advection equations is derived and analyzed, where the finite element method is constructed by using spline. A proper spline subspace is discussed for satisfying boundary conditions. Meanwhile, in order to get more accuracy solutions, spline method is connected with finite element method. Furthermore, the stability and convergence are discussed rigorously. Two numerical experiments are also presented to verify the theoretical analysis.

Topological indices of discrete molecular structure
Muhammad Aamer Rashid, Sarfraz Ahmad, Muhammad Kamran Siddiqui, Juan L. G. Guirao and Najma Abdul Rehman
2021, 14(7): 2487-2495 doi: 10.3934/dcdss.2020418 +[Abstract](1763) +[HTML](389) +[PDF](433.64KB)

Topological indices defined on molecular structures can help researchers better understand the physical features, chemical reactivity, and biological activity. Thus, the study of the topological indices on chemical structure of chemical materials and drugs can make up for lack of chemical experiments and can provide a theoretical basis for the manufacturing of drugs and chemical materials. In this paper, we focus on the family of smart polymer which is widely used in anticancer drugs manufacturing. In chemical graph theory, a topological index is a numerical representation of a chemical structure which correlates certain physico-chemical characteristics of underlying chemical compounds e.g., boiling point and melting point. More preciously, we focus on the family of smart polymer which is widely used in anticancer drugs manufacturing, and computed exact results for degree based topological indices.

Darcy-Forchheimer relation in Magnetohydrodynamic Jeffrey nanofluid flow over stretching surface
Ghulam Rasool, Anum Shafiq and Hülya Durur
2021, 14(7): 2497-2515 doi: 10.3934/dcdss.2020399 +[Abstract](2164) +[HTML](673) +[PDF](982.49KB)

Present article aims to investigate the heat and mass transfer developments in boundary layer Jeffery nanofluid flow via Darcy-Forchheimer relation over a stretching surface. A viscous Jeffery naonfluid saturates the porous medium under Darcy-Forchheimer relation. A variable magnetic effect normal to the flow direction is applied to reinforce the electro-magnetic conductivity of the nanofluid. However, small magnetic Reynolds is considered to dismiss the induced magnetic influence. The so-formulated set of governing equations is converted into set of nonlinear ODEs using transformations. Homotopy approach is implemented for convergent relations of velocity field, temperature distribution and the concentration of nanoparticles. Impact of assorted fluid parameters such as local inertial force, Porosity factor, Lewis and Prandtl factors, Brownian diffusion and Thermophoresis on the flow profiles is analyzed diagrammatically. The drag force (skin-friction) and heat-mass flux is especially analyzed through numerical information compiled in tabular form. It has been noticed that the inertial force and porosity factor result in decline of momentum boundary layer but, the scenario is opposite for thermal profile and solute boundary layer. The concentration of nanoparticles increases with increased porosity and inertial effect however, a significant reduction is detected in mass flux.

Marangoni forced convective Casson type nanofluid flow in the presence of Lorentz force generated by Riga plate
Ghulam Rasool, Anum Shafiq and Chaudry Masood Khalique
2021, 14(7): 2517-2533 doi: 10.3934/dcdss.2021059 +[Abstract](915) +[HTML](227) +[PDF](933.72KB)

The present communication aims to investigate Marangoni based convective Casson modeled nanofluid flow influenced by the presence of Lorentz forces instigated into the model by an aligned array of magnets in the form of Riga pattern. The exponentially decaying Lorentz force is considered using the Grinberg term. On the liquid - gas or liquid - liquid interface, a realistic temperature and concentration distribution is considered with the assumption that temperature and concentration distributions are variable functions of \begin{document}$ x $\end{document}. The set of so-formulated governing problems under the umbrella of Navier Stokes equations is transformed into nonlinear ODEs using suitable transformations. Homotopy approach is implemented to achieve convergent series solutions for the said problem. Influence of active fluid parameters such as Casson parameter, Brownian diffusion, Prandtl number, Thermophoresis and others on flow profiles is analyzed graphically. The fluctuation in local physical quantities such as heat and mass flux rates, is noticed to check the significance of current fluid model in many industrial as well as engineering procedures using nanofluids. The outcomes indicate that the effective Lorentz force assists the fluid motion that results in an augmented velocity profile with incremental values of modified Hartman number. Furthermore, incremental data of Casson parameter motivates significant reduction in velocity profile.

Optimal control strategy for an age-structured SIR endemic model
Hassan Tahir, Asaf Khan, Anwarud Din, Amir Khan and Gul Zaman
2021, 14(7): 2535-2555 doi: 10.3934/dcdss.2021054 +[Abstract](764) +[HTML](155) +[PDF](894.1KB)

In this article, we consider an age-structured SIR endemic model. The model is formulated from the available literature while adding some new assumptions. In order to control the infection, we consider vaccination as a control variable and a control problem is presented for further analysis. The method of weak derivatives and minimizing sequence argument are used for deriving necessary conditions and existence results. The desired criterion is achieved and sample simulations were presented which shows the effectiveness of the control.

Self-adaptive algorithm based on a posteriori analysis of the error applied to air quality forecasting using the finite volume method
Hatim Tayeq, Amal Bergam, Anouar El Harrak and Kenza Khomsi
2021, 14(7): 2557-2570 doi: 10.3934/dcdss.2020400 +[Abstract](3256) +[HTML](1404) +[PDF](1013.34KB)

In this work, we present a self-adaptive algorithm based on the techniques of the a posteriori estimates for the transport equation modeling the dispersion of pollutants in the atmospheric boundary layer at the local scale (industrial accident, urban air quality). The goal is to provide a powerful model for forecasting pollutants concentrations with better manipulation of available computing resources.

This analysis is based on a vertex-centered Finite Volume Method in space and an implicit Euler scheme in time. We apply and validate our model, using a self-adaptive algorithm, with real atmospheric data of the Grand Casablanca area (Morocco).

Existence and uniqueness results for a smoking model with determination and education in the frame of non-singular derivatives
Sümeyra Uçar
2021, 14(7): 2571-2589 doi: 10.3934/dcdss.2020178 +[Abstract](736) +[HTML](88) +[PDF](648.06KB)

These days, it is widely known that smoking causes numerous diseases, as well as resulting in many avoidable losses of life globally, and therefore encumbers the society with enormous unnecessary burdens. The aim of this study is to examine in-depth a smoking model that is mainly influenced by determination and educational actions via CF and AB derivatives. For both fractional order models, the fixed point method is used, which allows us to follow the proof of existence and the results of uniqueness. The effective properties of the above-mentioned fractional models are theoretically exhibited, their results are confirmed by numerical graphs by various fractional orders.

Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation
Asif Yokus and Mehmet Yavuz
2021, 14(7): 2591-2606 doi: 10.3934/dcdss.2020258 +[Abstract](1354) +[HTML](129) +[PDF](2945.19KB)

In this paper, we investigate some analytical, numerical and approximate analytical methods by considering time-fractional nonlinear Burger–Fisher equation (FBFE). (1/G\begin{document}$ ' $\end{document})-expansion method, finite difference method (FDM) and Laplace perturbation method (LPM) are considered to solve the FBFE. Firstly, we obtain the analytical solution of the mentioned problem via (1/G\begin{document}$ ' $\end{document})-expansion method. Also, we compare the numerical method solutions and point out which method is more effective and accurate. We study truncation error, convergence, Von Neumann's stability principle and analysis of linear stability of the FDM. Moreover, we investigate the \begin{document}$ L_{2} $\end{document} and \begin{document}$ L_\infty $\end{document} norm errors for the FDM. According to the results of this study, it can be concluded that the finite difference method has a lower error level than the Laplace perturbation method. Nonetheless, both of these methods are totally settlement in obtaining efficient results of fractional order differential equations.

2021 Impact Factor: 1.865
5 Year Impact Factor: 1.622
2021 CiteScore: 3.6

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