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Analytical effects of viscous dissipation on time-dependent Casson nanofluid (CN) on a stretching surface are the focus of the current research. We used the proper transformations to change a set of PDEs into NODEs. This scheme is the homotopy analysis method (HAM). Graphs are used to plan the effects of various parameters on energy and momentum profiles, such as the magnetic parameter, nanoparticles volume friction unsteady parameter, parandtl number thermophoresis parameter, and Eckert number. In order to simulate and assess the structures of flow features, such as temperature and velocity profiles, in response to changes in developing factors, a physical description is utilized.

This study examines the impact of viscous dissipation past a fluctuating disk on the three-dimensional unsteady incompressible viscous flow of water-based nanofluids. The problem’s mathematical modeling entails the formulation of continuity, momentum, and energy equations—three fundamental nonlinear governing equations. The proper transformations were conducted in order to convert a collection of partial differential equations (PDEs) to nonlinear ordinary differential equations (NODEs). The obtained governing equations are then solved analytically using the homotopy analysis method (HAM). The study looks into the effects of many factors. To illustrate the influence of these characteristics, the study displays temperature filed and velocity filed as graphical representations. It is anticipated that this work will progress the field of condensed nanostructure and nanomaterials, providing new directions for future investigation in applications pertaining to heat transfer enhancement.

In this paper, we theoretically investigate the bifurcations and exact traveling wave solutions of a weakly nonlinear Schrödinger equation that includes higher-order dispersion effects. By analyzing the phase portraits, we predict the exact analytically solutions for the model equation. The predicted periodic waves and solitary waves of the model are determined using the Riccati sub-ODE and elliptic expansion methods, respectively. The proposed computational approaches in comparison to traditional computational techniques, offer powerful improvements in accuracy, efficiency, and the ability to identify a wide range of solitons solutions. Through linear stability analysis, we derive an analytical expression for the instability gain and analyze its main characteristics. Finally, we assess the robustness and stability of the solitary wave solutions through numerical simulations and compare the accuracy of the analytical and numerical results. The outcomes may be used as a guide for the production of solitons and ultra-short optical pulses for potential applications in the telecommunication domain.

In this article, the two-dimensional micropolar rotating nanofluid between two parallel plates under Hall's influence can be solved using the Hermite wavelet technique (HWT). The nanofluid flow between two parallel plates is taken under the influence of the Hall current. The flow of micropolar nanofluid has been assumed in a steady state. We obtain a coupled, highly nonlinear ordinary differential equations (ODEs) from the fundamental governing equations via appropriate similarity variables. The behavior of velocity, temperature, concentration, and micro rotation are computationally discussed concerning the various governing parameters. Our outcome illustrates that the HWT is immensely capable and accessible for finding solutions to this type of coupled nonlinear ODEs. The results are in outstanding agreement for coupled nonlinear ODEs in engineering applications.

This work investigates new exact solutions within a unique (2+1)-dimensional problem by combining the Hirota bilinear forms of the Kadomtsev–Petviashvili and the Shallow Water Wave equations. We obtain resonant Y-type and X-type soliton, breather, and lump waves by introducing novel limitations on the N -soliton. Additionally, it generates various types of solutions (bulk-soliton, fissionable soliton-lump, breather-lump, and soliton-breather-lump) based on velocity and module resonance conditions. Furthermore, we investigate the lump wave's paths interacting with the waves prior to and following collision using the long-wave limit approach. We show that the lump wave either avoids collision with other waves or maintains a persistent state of collision with them by applying additional limitations. Specifically, we provide a series of figures depicting all solutions, comprehensively capturing their dynamical behavior and interactions. • Exact solutions for the (2+1)-D K–P- Shallow Water Wave equation are presented. • Various wave interactions like Y and X-type solitons, breather waves, and lump waves are presented. • The study is supported by both theoretical analysis and numerical simulations. • The obtained results are given by 2D and 3D figures. [ABSTRACT FROM AUTHOR]

In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

AbstractIn this work, a powerful approach is presented to solve the nonlinear wave equation (NWE): utt(x, t) − u(x, t)uxx(x, t) = g(x, t), where cis a positive constant. Firstly, a discretization is implemented on the original equation. Then, a geometric technique is applied to obtain the approximate solutions. Some examples are displayed to prove the power and capability of the offered approach to work out the wave problem.

The dynamics of waves in a two-layer fluid system using a novel proportional derivative, referred to as the (\omega)-Derivative, which is introduced here, are probed. Exact solutions for the gRE-(\omega) D are derived by implementing a unified method. The results reveal many patterns of rogue waves (RWs), solitons, and complex waves with tunneling. The onset of RWs plays a crucial role in the dynamics of water waves in oceans and overseas, and have significant implications for research in fluid physics. The stability of the steady state solution (SSS) is analyzed, which is found to be unstable when the coefficient of the highest-order dispersion exceeds a critical value. We think that this work contributes new insights into the patterns of rogue waves, which may have impact on the waves dynamics. • A new ω -derivative is introduced. • Ramani equation with ω -time derivative is studied. • Multiple waves geometric structures are revealed, among them rogue waves in top pattern, rogue waves with breathers back. • The stability of the steady state solution is analyzed. [ABSTRACT FROM AUTHOR]

The goal of this research is to explore the HIV infection model through Caputo derivative in considerable detail. We employ a non-local fractional derivative having singular kernel to thoroughly evaluate the dynamics of the HIV infection model. Also, we present some important theoretical results and main features of the fractional disease model addressed. The dynamic behavior of the HIV infection is investigated by a suitable numerical scheme based on the Caputo fractional operator. Furthermore, the non-linear least-squares curve fitting procedure is utilized for estimating the parameters of the proposed model with real-world data. On the other hand, it is worth noting that during the fractionalization process, we take into account dimensional consistency.

Highlights • The model of transmission of vector-borne diseases and cure is studied. • The Atangana–Baleanu fractional operator in caputo sense is used to study the model. • Existence and uniqueness are shown and numerical simulations of the fractional model are derived. Abstract The model of transmission dynamics of vector-borne diseases with vertical transmission and cure within a targeted population is extended to the concept of fractional differentiation and integration with non-local and non-singular fading memory introduced. The effect of vertical transmission and cure rate on the basic reproduction number is shown. The Atangana–Baleanu fractional operator in caputo sense (ABC) with non-singular and non-local kernels is used to study the model. The existence and uniqueness of solutions are investigated using the Picard–Lindel method. Ultimately, for illustrating the acquired results, we perform some numerical simulations and show graphically to observe the impact of the arbitrary order derivative. It is expected that the proposed model will show better approximation than the classical model established before. [ABSTRACT FROM AUTHOR]

Highlights • Some nonlinear partial differential equations with new fractional derivative are investigated. • The existence and uniqueness analysis of the solution are reported. • The numerical results are obtained by using an iterative method. • Efficincy of Atangana–Baleanu fractional derivative over well known Caputo–Fabrizio fractional derivative is illustrated. Abstract In this work, the efficiency of the Atangana–Baleanu (AB) derivative over Caputo–Fabrizio (CF) to some nonlinear partial differential equation is presented. The considered equations are Rosenou–Haynam equation (RHE) and a class of mKdV (CmKdV) equation. The effective and efficient technique called the fractional homotopy perturbation transform method (FHPTM) is applied for the investigation of the governing equations. [ABSTRACT FROM AUTHOR]

In this work, we investigate the Lie symmetry analysis, exact solutions and conservation laws (Cls) to the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGDK) equation with Riemann-Liouville (RL) derivative. The time fractional CDGDK is reduced to nonlinear ordinary differential equation (ODE) of fractional order. New exact traveling wave solutions for the time fractional CDGDK are obtained by fractional sub-equation method. In the reduced equation, the derivative is in Erdelyi-Kober (EK) sense. Ibragimov’s nonlocal conservation method is applied to construct Cls for time fractional CDGDK. [ABSTRACT FROM AUTHOR]

•The Kundu–Eckhaus (KE) equation is studied.•Optical solitons are obtained by using modified tanh–coth and Jacobi elliptic function expansion methods.•Nonlinear self-adjointness for the KE equation is investigated.•Conservation laws are constructed via nonlinear self-adjointness.

The current paper raises notice of a novel approach known as the “heir-equations method.” Nucci, for the first time in the literature, discovered that iterating the nonclassical symmetry method to generate new nonlinear equations secures the Lie point symmetry properties of the given equation, or, in other words, inherits them. The technique that corresponds to non-classical symmetries is particularly powerful in obtaining new solutions to evolutionary-type equations. In this article, new solutions and nonclassic symmetries of the coefficient-variable forced Burger equation are obtained by the heir-equations method.

Nonlinear evolution equations (NLEEs) are extensively used to establish the elementary propositions of natural circumstances. In this work, we study the Konopelchenko–Dubrovsky (KD) equation which depicts non-linear waves in mathematical physics with weak dispersion. The considered model is investigated using the combination of generalized exponential rational function (GERF) method and dynamical system method. The GERF method is utilized to generate closed-form invariant solutions to the (2+1)-dimensional KD model in terms of trigonometric, hyperbolic, and exponential forms with the assistance of symbolic computations. Moreover, 3D, 2D combined line graph and their contour graphics are displayed to depict the behavior of obtained solitary wave solutions. The model is observed to have multiple soliton profiles, kink-wave profiles, and periodic oscillating nonlinear waves. These generated solutions have never been published in the literature. All the newly generated soliton solutions are checked by putting them back into the associated system with the soft computation via Wolfram Mathematica. Moreover, the system is converted into a planer dynamical system using a certain transformation and the analysis of bifurcation is examined. Furthermore, the quasi-periodic solution is investigated numerically for the perturbed system by inserting definite periodic forces into the considered model. With regard to the parameter of the perturbed model, two-dimensional and three-dimensional phase portraits are plotted.

The current work deals with the classical and non-classical Lie group analysis for the fractional type system of nonlinear Chen–Lee–Liu (CLL) equations. The invariance surface condition is imposed to this system of fractional differential equations to find non-classical generators. Non-classical and classical Lie symmetry analysis are used to similarity reductions of this system. Corresponding exact solutions are obtained for the extracted generators in both classical and non-classical senses. Convergence of the obtained solutions when fractional order tends to the integer one, is graphically checked. Finally, the conservation laws are investigated for the system of time-fractional CLL equations, specially for the non-classical vector fields.

The connection of Zagreb polynomials and Zagreb indices to chemical graph theory is a bifurcation of mathematical chemistry, which has had a crucial influence on the development of chemical sciences. Nowadays, the study of topological indices has become a vast effective research area in chemical graph theory. In this article, we add up eight different Zagreb polynomials for the Silicate Network and Silicate Chain Network. From these Zagreb polynomials, we catch up on degree-based Zagreb indices. We also provide a graphical representation of the outcome that describes the dependence of topological indices on the given parameters of polynomial structure.

Considering the critical role of electricity in the industrial and economic infrastructures and the large investments in this sector, it is necessary to correctly utilize and maintain this great industry. Further, due to the increasing prices of energy, environmental damages, and the limitation of non-renewable energy sources, it is increasingly important to maximize the performance of energy production and transformation systems. This study aims to model and analyze the economic exergy of the 60 MW heat and power generation power plant in the Esfahan steel company, as the installation has never been modeled or optimized. After 50 years of operation, the results of modeling the steam power plant and exergoeconomic analysis can lead to optimal operating conditions and be of benefit to the stakeholders. The findings indicate that the condenser, or rather cooling tower is where the maximum loss of energy occurs, whereas the boiler has the most exergy loss while the turbine ranks next in exergy loss. Increasing the load ratio increases the cycle efficiency, whereas maximum efficiency and the least heat rate occur at full load. However, increased load also increases energy/exergy loss. Increasing the temperature of the feedwater plus decreasing the condenser pressure can reduce energy/exergy loss.

The interfacial dynamics of a system are significant in understanding the system’s behavior. One of the special models of these systems which naturally arises in material sciences is the phase separation (or spinodal decomposition) of binary alloys. A nonlinear evolution equation known as the Cahn–Hilliard equation is used to establish the mathematical modeling of a binary alloy for phase separation. Apart from trivial solutions, the Cahn–Hilliard equation is a fourth-order nonlinear equation without an analytical solution. This study investigates a second-order splitting finite difference scheme based on the 2D Crank–Nicolson technique to approximate the solution of the 2D Cahn–Hilliard problem with Neumann boundary conditions. Using the inherent error estimation of the 2D Crank–Nicolson method, we have shown that the proposed scheme is second-order accuracy and we have proved that the scheme has a unique solution. In addition, we demonstrated that the suggested technique retains mass conservation while decreasing total energy. Furthermore, we chose two numerical examples, one with a special initial value and the other with a random initial value, to confirm the accuracy of the proposed approach. These numerical experiments display physical features of the Cahn–Hilliard model such as coarsening dynamics and spinodal decomposition obtained by experimental studies of binary alloys.

This work is devoted to the time-fractional differential equations with the regularized Prabhakar derivative and their analytical solutions. We generalize the invariant subspace method to find the exact solutions of such equations. Then, we apply this method to obtain the exact solutions of different time-fractional nonlinear differential equations including the regularized Prabhakar derivative.

In this paper, we use the reproducing kernel Hilbert space method for solving a boundary value problem for the second order Bratu’s differential equation. Convergence analysis of presented method is discussed. The numerical approximations to the exact solution are computed and compared with other existing methods. Our presented method produces more accurate results in comparison with those obtained by Adomian decomposition, Laplace decomposition, B-spline, non-polynomial spline and Lie-group shooting methods. Our yardstick is absolute error. The comparison of the results with exact ones is made to confirm the validity and efficiency.

In this article, we present the Homotopy Perturbation Method (Shortly HPM) for obtaining the numerical solutions of the Korteweg‐de Vries Burgers (KdVB) equation. The series solutions are developed and the reccurance relations are given explicity. The initial approximation can be freely chosen with possibly unknown constants which can be determined by imposing the boundary and initial conditions. The results reveal that HPM is very simple and effective. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Abstract: In this paper, we obtained integrable evolution equation from binormal motions of curves in the 3-dimensional Euclidean spaceAfterwards, this equation is solved by sine–cosine method and compacton solutions are obtained. As a result, abundant new compactons solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave and periodic wave solutions are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, the nonlinear dispersive Zakharov–Kuznetsov equation was solved by using the sine–cosine method. As a result, new solitary pattern, solitary wave and singular solitary wave solutions were found. [Copyright &y& Elsevier]

Abstract: In this paper, the homotopy analysis method (HAM) has been used to evaluate the efficiency of straight fins with temperature-dependent thermal conductivity and to determine the temperature distribution within the fin. The fin efficieny of the straight fins with temperature-dependent thermal conductivity has been obtained as a function of thermo-geometric fin parameter. It has been observed that the thermal conductivity parameter has a strong influence over the fin efficiency. The series solution is developed and the reccurance relations is given. Comparison of the results with those of the homotopy perturbation method (HPM) and the Adomian decomposition method (ADM), has led us to significant consequences. The analytic solution of the problem is obtained by using the HAM. The HAM contains the auxiliary parameter , which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter , we can obtain reasonable solution for large values of and . [Copyright &y& Elsevier]

Abstract: In this paper, we establish exact special solutions for the nonlinearly dispersive Boussinesq equations. A new extended sn–cn method is used for obtaining compacton, solitary patterns, solitary wave, periodic wave and Jacobi elliptic function solutions for this equation with minimal calculations. As a result, abundant new compactons:solitons with the absence of infinite wings, solitary pattern solutions having infinite slopes or cups, solitary wave, periodic wave solutions and Jacobi elliptic function solutions are obtained. [Copyright &y& Elsevier]

Abstract: A scheme is developed for the numerical study of the Korteweg-de Vries (KdV) and the modified Korteweg-de Vries (mKdV) equations with initial conditions by a variational approach. The exact and numerical solutions obtained by variational iteration method are compared with those obtained by Adomian decomposition method. The comparison shows that the obtained solutions are in excellent agreement. [Copyright &y& Elsevier]

In this paper, we study the modified decomposition method (MDM) for solving nonlinear twopoint boundary value problems (BVPs) and show numerical experiments. The modified form of the Adomian decomposition method which is more fast and accurate than the standard decomposition method (SDM) was introduced by Wazwaz. In addition, we will compare the performance of the MDM and the new nonlinear shooting method applied to the solutions of nonlinear two-point BVPs. The comparison shows that the MDM is reliable, efficient and easy for solving the nonlinear twopoint BVPs.

Abstract: In this paper, the nonlinear dispersive Zakharov–Kuznetsov ZK(m, n, k) equations are solved exactly by using the Adomian decomposition method. The two special cases, ZK(2,2,2) and ZK(3,3,3), are chosen to illustrate the concrete scheme of the decomposition method in ZK(m, n, k) equations. General formulas for the solutions of ZK(m, n, k) equations are established. [Copyright &y& Elsevier]

Abstract: The special exact solutions of the modified nonlinearly Klein–Gordon-type equations mKG(m, n, k) are investigated by using some ansatze. Thus, we obtain new soliton solutions with compact support and solitary pattern solutions having infinite slopes or cusps, solitary wave and periodic solutions are obtained. The properties of the mKG(m, n, k) equations are shown in some figures. [Copyright &y& Elsevier]

Abstract: In this paper, the nonlinear Volterra–Fredholm integral equations are solved by using the modified decomposition method (MDM). The approximate solution of this equation is calculated in the form of a series with easily computable components. The accuracy of the proposed numerical scheme is examined by comparison with other analytical and numerical results. Two test problems are presented to illustrate the reliability and the performance of the modified decomposition method. [Copyright &y& Elsevier]

Abstract: In this paper, we establish exact solutions for the K(k, n) equations. A new sn–cn method is used for obtaining traveling wave solutions for this equation with minimal calculations. As a result, abundant new compactons: solitons with the absence of infinite wings, solitary wave and periodic wave solutions of this equation is obtained. The properties of the K(k, n) equations are shown in figures. [Copyright &y& Elsevier]

Abstract: In this paper, we establish exact solutions for the R(m, n) equations by using a new sc–nc method. As a result, abundant new exact solitary pattern, solitary wave, periodic wave and singular solitary wave solutions of this equation are obtained with minimal calculations. The new solitary pattern solutions have infinite slopes or cusps, solitary wave and singular solitary wave solutions. [Copyright &y& Elsevier]

In this paper, we find travelling wave solutions of some nonlinear partial differential equations (PDEs) by using the extended tanh-function method. Some illustrative equations are investigated by this method and new travelling wave solutions are found. In addition, the properties of these nonlinear PDEs are shown with some figures.

In this paper we investigate exact solutions of a modified form of fifth-order Korteweg-de Vrieslike equations by using two direct methods. Thus we get new compacton solutions having infinite wings or tails. In addition, new periodic and singular periodic wave solutions are obtained.

In this paper we shall derive new solitary pattern solutions of the nonlinear dispersive KdV equation (shortly called K(n,m,k) ) by use of simple methods. The properties of these K(n,m,k) equations are shown in figures.

In pursuit of improved thermal transportation, the slip flow of Casson nanofluid is considered in the existence of an inclined magnetic field and radiative heat flux flow over a non-linear stretching sheet. The viscosity of the fluid is considered as a function of temperature along with the convective thermal boundary condition. Numerical solutions are obtained via Runge-Kutta along with the shooting technique method for the chosen boundary values problem. To see the physical insights of the problem, some graphs are plotted for various flow and embedded parameters on temperature function, micro-organism distribution, velocity, and volume fraction of nanoparticles. A decline is observed in the velocity and the temperature for Casson fluid. Thermophoresis and Brownian motion incremented the temperature profile. It is also found that thermal transportation can be enhanced in the presence of nanoparticles and the bioconvection of microorganisms. Present results are useful in the various sectors of engineering and for heat exchangers working in various technological processors. The main findings of the problem are validated and compared with those in the existing literature as a limiting case.

The impacts of chemical reaction and suction/injection in Williamson fluid flow along a porous stretching wedge are discussed in the present paper. Recently, a lot of numerical and theoretical studies are accessible for illustrating the chemical reaction impact on non-Newtonian fluids with different geometries and conditions. Considering this fact, we inspect the heat transport behavior of Williamson fluid due to a stretching porous wedge with suction or injection. The governing PDEs are converted into ODEs with reliable similarity transformation. These ODEs are solved numerically by BVP4C based in MATLAB system, and the assessment of outcomes for the validation tenacity is presented in Table. Combined plots are sketched to discern the impact of dominant sundry parameters on the flow fields. Along with them, the Sherwood number, skin friction factor, and the rate of heat transfer are also bestowed in graphs.

In this manuscript, the aim is to prove a multiple fixed point (FP) result for partially ordered s-distance spaces under θ,ϕ,ψ-type weak contractive condition. The result will generalize some well-known results in literature such as coupled FP (Guo and Lakshmikantham, 1987), triple fixed point (Berinde and Borcut, 2011), and quadruple FP results (Karapinar, 2011). Moreover, to validate the result, an application for the existence of solution of a system of integral equations is also provided.

• The fractional lana fever infectious ailment model is studied. • The Existence and uniqueness and stability analysis are presented. • The obtained results are given by figures. This study provides a deep insight into the complex process of the lana fever infectious disease model by employing the advantages of a non-local fractional operator called Caputo derivative. The fractional-type lana fever model possessing memory effect and hereditary properties allows us to comprehend the dynamics of the disease transmission in detail. For this purpose, we introduce some crucial theoretical and numerical results for the suggested disease model with the aid of the Caputo fractional derivative. The existence and uniqueness of the solution of lana fever system are shown through the fixed point theorem. Moreover, we find the most sensitive parameters under the forward sensitivity index in relation to reproduction number R 0. Also, some basic properties of the lana fever model such as virus-free equilibrium point, reproduction number, virus endemic equilibrium point, local and global stability are presented by using the Caputo derivative. On the other hand, further analysis of the fractional version of the lana fever model shows that the disease-free equilibrium is locally asymptotically stable when R 0 < 1 , and also the endemic equilibrium of the model under investigation is globally asymptotically stable when R 0 > 1. Additionally, numerical results are presented to observe the advantages of the non-local fractional differential operator of Caputo. [ABSTRACT FROM AUTHOR]

• The nonlinear chaotic complex system are studied. • The ultimate bound of the nonlinear chaotic complex systems are presented. • The obtained solutions are presented by 2D figures. The competitive modes for a nonlinear chaotic complex system are studied in this paper. In hyperchaotic, chaotic, and periodic cases, we examined competitive modes that are a tool for detecting chaos in a system. Also, using an analytical method and Lagrange optimization, we were able to calculate the ultimate bound of the nonlinear chaotic complex systems. We have presented is simpler and more accurate than other methods that implicitly calculate the ultimate bound. The estimation of the explicit ultimate bound can be used to study chaos control and chaos synchronization. Numerical simulations illustrate the analytical results. [ABSTRACT FROM AUTHOR]

• The fractional variable-order mathematical model for hepatitis C having various types of the virus genome is studied. • The numerical methodology proposed with the interpolation of the Lagrange polynomial is established. • The equilibrium points and the basic reproductive number R 0 are presented. • The obtained approximate solutions are presented by 2D figures. The basic objective of this work is to present and study an effective and cost-efficient numerical technique for analyzing the fractional variable order mathematical model for hepatitis C having various types of the virus genome. Here, we give this model via the most commonly used non-singular Caputo–Fabrizio fractional derivative (CF) depending upon the exponential-kernel. The Hepatitis C virus (HCV) is an RNA virus that is single-stranded. The HCV genomes show substantial variability in sequence and are categorized into both types and subtypes. Those types in the range from 1 to 11 are recognized so far (every kind with a specific number of sub-kinds). It is reported that around 90 % of the isolated HCV infections occurring in Egypt goes into only a single subtype (4a). The numerical methodology proposed is established on the fractional calculus basic theorem with the interpolation of the Lagrange polynomial. The effectiveness of the proposed procedure are satisfied. The findings indicate that the technique considered is a simple and successful method for investigating the solution of such models. The 4th-order Runge–Kutta method (RK4), is utilized to compare the numerical solutions and show the excellent agreement that has been found out via applying the CFC-derivatives. The obtained solutions of the suggested model illustrate the simple usage and the effectiveness of the presented method and are in agreement with the characteristics of the virus. [ABSTRACT FROM AUTHOR]

• Consumption control has a crucial position for maintaining the balance between energy efficiency and energy demand. • Investigating the heating and cooling of buildings in many ways with the help of mathematical tools gains an advantage for energy-saving. • Energy can be reserved by acting efficient behaviors, procuring insulation, upgrading equipment, and so on. In the current study, the temperature dynamics of a building are proposed by means of some non-local singular fractional-order operators. In order to examine this model which has a crucial place in daily life, we utilize the Laplace transform. Accordingly, new analytical solutions including one, two, or three parameters are displayed to better control the non-renewable resources. One of the most remarkable features making the aforementioned model important is that it has different non-integer parameters α , ρ , and γ , which enable us to approach maximum energy efficiency for buildings. Moreover, simulation analysis is carried out to observe the mobility of the solution curve in line with our objective. [ABSTRACT FROM AUTHOR]

In this article, we consider an extensive class of monotone nonexpansive mappings. We use S-iteration to approximate the fixed point for monotone total asymptotically nonexpansive mappings in the settings of modular function space.