Narrow Results

This research proposes a modified Crank–Nicolson finite difference scheme with local fractional derivatives to approximate the solutions of local fractional LWR traffic flow model. The stability and consistency of the scheme are examined. Further, convergence of the scheme is assured by using Lax’s equivalence theorem. Some exemplary instances are discussed along with their simulations to validate the proposed method. The obtained numerical solutions show the dynamical evolution of traffic density with respect to time and space. The results derived using the proposed numerical scheme establish that they are quite effective in obtaining the numerical solution to the fractal vehicular traffic flow problem.

This investigation predicts the assessment of mass and heat transfer due to Burgers nanofluid. The investigation is further supported by triple diffusion flow. Variable thermal conductivity is accounted to endorse the thermal flow. In the heat equation, the extension is suggested by utilizing the external heat source and nonlinear radiated phenomenon. Computations for modeled problems are achieved by the shooting method. The fundamental role of parameters governing flow is noticed which is physically attributed. It is observed that the heat transfer rate is enhanced due to the modified Dufour number. The solutal concentration declined for the regular Lewis number. Furthermore, the nanoparticles concentration reduces due to the nano-Lewis number. The depicted results convey novel applications in chemical processes, cooling control systems, refrigeration, solar energy, extrusion processes, etc.

We propose a thermal lattice Boltzmann model to study gaseous flow in microcavities. The model relies on the use of a finite difference scheme to solve the set of evolution equations. By adopting diffuse reflection boundary conditions to deal with flows in the slip regime, we study the micro-Couette flow in order to select the best numerical scheme in terms of accuracy. The scheme based on flux limiters is then used to simulate a micro-lid-driven cavity flow by using an efficient and parallel implementation. The numerical results are in very good agreement with the available results recovered with different methods.

This motivating analysis aims to present the thermal mechanism for mixed convection flow of Reiner–Philippoff nanofluid with assessment of entropy generation. The thermal performances of nanomaterials have been modified by utilizing the nonuniform heat source/sink, Ohmic dissipations and thermal radiation consequences. The assumed surface is assumed to be porous with non-Darcian porous medium. The modified Cattaneo–Christov relations are followed to modify the mass and heat equations. The invoking of similarity variables results in differential equations in nonlinear and coupled form. A MATLAB-based shooting algorithm is employed to access the numerical simulations. The physical aspect of thermal model is graphically addressed for endorsed flow parameters. The importance of entropy generation is visualized with associated mathematical relations and physical explanations. The numerical values are obtained for the assessment of heat and mass transfer phenomenon.

The study of multiphase flows gained much importance because of its extensive applications in nature and industry. These flows possess two or more thermodynamic phases, for example, one component phase (e.g., water vapors and water flow) or several components phase (e.g., water and oil flow). The most common example of multiphase flow in the context of the oil industry is petroleum. Further blood flow, porous structures, fluidized bed, bubbly flow in nuclear reactors, and fiber suspension in the paper industry are some significant examples of multiphase flows. In this paper, we considered the Couette flow of non-Newtonian (couple stress) fluid with variable magnetic field and thermal conductivity effects between parallel walls of the channel. The upper wall of the channel is in constant motion while the lower wall is in a fixed position. The variable viscosity effects with the suspension of hafnium particles are also discussed by taking Vogel’s viscosity case. The shooting method based on the R–K method is applied to obtain the numerical solution of the current problem. A comparison between Newtonian and non-Newtonian fluids is presented by sketching graphs. The variations in flow and temperature of fluid against various involved factors, including variable viscosity, wall temperature, thermal radiations, variable magnetic field, and thermal conductivity are sketched and also physically described. It is observed that variable viscosity parameter elevated both velocity and temperature profiles while wall temperature parameter decelerated both fields. Further, noticed that the variable thermal conductivity and variable magnetic field impede the velocity of the fluid and also retarded the temperature field. Our attempt is not just useful to investigate the mechanical and industrial multiphase flows but also delivers important results to fill the gap in the existing literature.

In nanotechnology, the nanofluids are decomposition of base materials and nanoparticles where the nanoparticles are immersed in base liquid. The utilization of such nanoparticles into base liquids can significantly enhance the thermal features of resulting materials which involve applications in various industrial and technological processes. While studying the rheological features of non-Newtonian fluids, the constant viscosity assumptions are followed in many investigations. However, by considering the viscosity as a temperature-dependent is quite useful to improve the heating processes along with nanoparticles. Keeping such motivations in mind, this investigation reports the temperature-dependent viscosity and variable heat-dependent conductivity in bioconvection flow of couple stress nanoparticles encountered by a moving surface. The famous Reynolds exponential viscosity model is used to deploy the relations for temperature-dependent viscosity. Moreover, the activation energy and higher order slip (Wu’s slip) are also elaborated to make this investigation more novel and unique. The emerging flow equations for governing flow problem are formulated which are altered into non-dimensional forms. The numerical simulations with applications of Runge–Kutta fourth–order algorithm are focused to obtain the desired solution. Before analyzing the significant physical features of various parameters, the confirmation of solution is done by comparing the results with already reported investigations as limiting cases. The results are graphically elaborated with relevant physical consequences. Various plots for velocity, temperature, concentration, wall shear stress, local Nusselt number, local Sherwood number and motile density numbers are prepared.

The study focuses on the flow of hybrid nanofluid, induced by magnetic and radiation effects, across an exponentially stretched sheet. The research examines the impact of temperature-dependent properties of the hybrid nanofluid on the sheet. Water is used as the base fluid, and SWCNT and MWCNT are employed as nanoparticles. The study includes a discussion of the Yamada–Ota, Xue and Tiwari–Das models of hybrid nanofluids. The governing system of flow is presented mathematically, and boundary layer approximations are used to reduce differential equations. The differential equations are transformed into dimensionless ordinary differential equations (ODEs) by using transformations. The dimensionless system of equations is then solved numerically. The results of the flow model are offered in tabular and graphical forms. We observed that Tiwari–Das model of hybrid nanofluid achieved more heat transfer and friction factor values when compared to other models of Xue and Yamada–Ota models of hybrid nanofluid. Temperature curves are noted to be enhanced by enlargement in the nano-concentration factor. If the nano-concentration increased in the fluid which boosted the thermal conductivity of the liquid, then as a result, the temperature of fluid enhanced at surface.

Owing to the growing interest of bioconvection flow of nanomaterials, many investigations on this topic have been performed, especially in this decade. The bioconvection flow of nanofluid includes some novel significance in era of biotechnology and bio-engineering like bio-fuels, microbial enhanced oil recovery, enzymes, pharmaceutical applications, petroleum engineering, etc. The current analysis aims to explore the various thermal properties of Sutterby nanofluid over rotating and stretchable disks with external consequences of variable thermal conductivity, heat absorption/generation consequences, activation energy and thermal radiation. The considered flow problem is changed into dimensionless form with convenient variables. The numerical structure for the obtained non-dimensional equations is numerically accessed with built-in shooting technique. The consequences of various physical parameters are observed for enhancement of velocity, temperature, concentration and motile microorganism. It is noted that both axial and tangential velocity components decrease with Reynolds number and buoyancy ratio parameter. The nanofluid concentration improves with activation energy and concentration Biot number. Moreover, an improved microorganisms profile is noticed with microorganism Biot number and bioconvection Rayleigh number.

The infection of coronavirus (COVID-19) is a dangerous and life-threatening disease which spread to almost all parts of the globe. We present a mathematical model for the transmission of COVID-19 with vaccination effects. The basic properties of fractional calculus are presented for the inspection of the model. We calculate the equilibria of the model and determined the reproduction number ${{ℛ}}_0$. Local asymptotic stability conditions for the disease-free are obtained which determines the conditions to stabilize the exponential spread of the disease. The nonlinear least-square procedure is utilized to parameterize the model from actual cases reported in Pakistan. By fixed point theory, we prove the existence of a unique solution. We also present numerical results to simulate virus transmission and compare the results with those of the Caputo derivative. We study the solution pathways of the COVID-19 system to provide effective control policies for the infection. Significant changes have been noticed by lowering the order of fractional derivative.

The application of the recently proposed integral and differential operators known as the fractal-fractional derivatives and integrals has opened doors to ongoing research in different fields of science, engineering, and technology. These operators are a convolution of the fractal derivative with the generalized Mittag-Leffler function with Delta-Dirac property, the power law, and the exponential decay law with Delta-Dirac property. In this paper, we aim to extend the work in the literature by applying these operators to a modified stretch–twist–fold (STF) flow based on the STF flow related to the motion of particles in fluids that naturally occur in the dynamo theorem. We want to capture the dynamical behavior of the modified STF flow under these operators. We will present the numerical schemes that can be used to solve these nonlinear systems of differential equations. We will also consider numerical simulations for different values of fractional order and fractal dimension.

We aim in this paper to study the effect of harvesting on predator–prey interaction in the case of prey herd behavior using a fractional-order model. Herd behavior has a crucial role in the surviving of species where it gave them the sufficient protection for the prey that perform it. The objective of using the fractional-order model is to model the memory effect measured by the order of the fractional derivative on the mutual interactions. Further, we aim to seek the effect of inner competition among the predators (also super-predators) on the evolution of the three species. For the mathematical results, we will show the local stability of the equilibria, and show the effect of memory rate and harvesting of the asymptotic behavior of the solution. Moreover, an efficient numerical scheme has been used to provide the numerical illustrations for our study.

This work is devoted to establish a numerical scheme for system of fractional-order differential equations (FODEs) with order $1<𝜗\le 2$. The scheme is established by using Bernstein polynomials (BPs). Based on the said materials, some operational matrices are formed. With the help of obtained operational matrices, the considered system is reduced to some algebraic system of equations. On using MATLAB-16, the system is then solved to get the required numerical solution for the proposed system. Several examples are treated with the help of the proposed method for numerical solutions. Further, error analysis is also recorded for different fractional orders and various scale levels. The mentioned results are displayed graphically. Comparison with exact solution at traditional order derivative is also given. It should be kept in mind that the proposed method does not require any kind of discretization or collocation. Also, there is no external parameter which controls the method. Due to these features, the proposed method is powerful and efficient for different classes of FODEs to compute their numerical solutions. The efficiency of the proposed method can be enhanced by increasing the scale level.

In this paper we study BSDEs arising from a special class of backward stochastic partial differential equations (BSPDEs) that is intimately related to utility maximization problems with respect to arbitrary utility functions. After providing existence and uniqueness we discuss the numerical realizability. Then we study utility maximization problems on incomplete financial markets whose dynamics are governed by continuous semimartingales. Adapting standard methods that solve the utility maximization problem using BSDEs, we give solutions for the portfolio optimization problem which involve the delivery of a liability at maturity. We illustrate our study by numerical simulations for selected examples. As a byproduct we prove existence of a solution to a very particular quadratic growth BSDE with unbounded terminal condition. This complements results on this topic obtained in Briand and Hu (2006, 2008) and Briand et al. (2007).

We propose and analyze a numerical scheme for the approximation of the solution for the stochastic 2D hydrodynamical type systems driven by non-Gaussian Lévy noise. We prove the convergence of the scheme which is a linear evolution equation with additive noise.

We prove an $L^2$-regularity result for the solutions of Forward Backward doubly stochastic differential equations (F-BDSDEs) under globally Lipschitz continuous assumptions on the coefficients. As an application of our result, we derive the rate of convergence in time for the (Euler time discretization-based) numerical scheme for F-BDSDEs proposed by Bachouch et al. (2016) under only globally Lipschitz continuous assumptions.

A linear modelling of aeroacoustic waves propagation is discussed. The first point is an existence and uniqueness theorem. But restrictive assumptions are required on the velocity of the flow. Then a counter example proves that they are necessary.

The infection of dengue is an intimidating vector-borne disease caused by a pathogenic agent that affects different temperature areas and brings many losses in human health and economy. Thus, it is valuable to identify the most influential parameters in the transmission process for the control of dengue to lessen these losses and to turn down the economic burden of dengue. In this research, we formulate the transmission phenomena of dengue infection with vaccination, treatment and reinfection via Atangana–Baleanu operator to thoroughly explore the intricate system of the disease. Furthermore, to come up with more realistic, dependable and valid results through fractional derivative rather than classical order derivative. The next-generation approach has been utilized to compute the basic reproduction number for the suggested fractional model, indicated by ${{ℛ}}_0$; moreover, we conducted sensitivity test of ${{ℛ}}_0$ to recognize and point out the role of parameters on ${{ℛ}}_0$. Our numerical results predict that the reproduction number of the system of dengue infection can be controlled by controlling the index of memory. The uniqueness and existence result has been proved for the solution of the system. A novel numerical method is presented to highlight the time series of dengue system. Eventually, we get numerical results for different assumptions of $\ell$ with specifying factors to conceptualize the effect of $\ell$ on the dynamics. It has been noted that the fractional-order derivative offers realistic, clear-cut and valid information about the dynamics of dengue fever. Moreover, we note through our analysis that the input parameters’ index of memory, biting rate, transmission probability and recruitment rate of mosquitos can be used as control parameter to lower the level of infection.

In this paper, we formulate the transmission phenomena of Hand–Foot–Mouth Disease (HFMD) through non-integer derivative. We interrogate the biological meaningful results of the recommended system of HFMD. The basic reproduction number is determined through next generation method and the impact of different parameters on the reproduction number is examined with the help of partial rank correlation coefficient (PRCC) technique. In addition, we concentrated on qualitative analysis and dynamical behavior of HFMD dynamics. Banach’s and Schaefer’s fixed-point theorems are used to analyze the uniqueness and existence of the solution of the proposed HFMD model. The HFMD system’s Ulam–Hyers stability has been confirmed to be sufficient. To highlight the impact of the parameters on the dynamics of HFMD, we performed several simulations through numerical scheme to conceptualize the transmission route of the infection. To be more specific, numerical simulations are used to visualize the effect of input parameters on the systems dynamics. We have shown the key input parameters of the system for the control of infection in the society.

Norovirus infection has been documented to have a significant economic impact in different regions of the world. Even though young children bear the greatest economic burden, older age groups in some locations have the highest costs per illness. Most of these costs result from lost production caused by acute illnesses. This viral infection causes inflammation of the intestines and stomach, also called stomach flu and food poisoning. Our research work constructs a new compartmental model of norovirus infection based on contaminated water and food contamination to conceptualize how norovirus spreads. The proposed dynamics of norovirus have been presented in the fractional Caputo framework. We present the Caputo operator’s rudimentary results for the system’s examination. By applying the fixed point theory to the system, the existence theory has been investigated. To inspect the solution pathways of norovirus infection, we introduce a novel numerical scheme to explore the dynamical behavior of the system. Finally, a numerical investigation has been done to show the impact of different factors of the system for the control of norovirus illness. We suggested the most critical factors of the system to the policymakers to control and prevent infection in the community.

We are interested in the existence results for second-order differential inclusions, involving finite number of unilateral constraints in an abstract framework. These constraints are described by a set-valued operator, more precisely a proximal normal cone to a time-dependent set. In order to prove these existence results, we study an extension of the numerical scheme introduced in [10] and prove a convergence result for this scheme.