Search

In the present time, thermal transportation in the colloidal suspensions under various scenario becomes an influential research direction due to their extensive applications. Therefore, investigation of thermal transport over a rotating disk under the impacts of thermal and velocity slip, imposed Lorentz forces and thermal radiation is conducted for multiple shape effects of the nanomaterial. The nanofluid model suspended by Al2O3, TiO2 and Cu nanomaterial is reduced in dimensionless version via similarity variables. After that, RK scheme is implemented and handle the model effectively. The outcomes of various parameters for the velocity, thermal transport, skin friction and local heat transport are sketched and explained broadly. It is examined that the heat transport for the nanofluids becomes dominant throughout the analysis in comparison with conventional liquid. The temperature of the nanofluids significantly enhances due to the velocity slip effects. Moreover, thermal radiation and the volumetric fraction of the nanomaterials favor the local heat transfer rate.

Abstract The dynamics of nanofluid by considering the role of imposed Lorentz forces, thermal radiations and velocity slip effects over a vertically convectively heated surface is a topic of huge interest. Therefore, the said study is conducted for Al2O3-H2O nanofluid. Mathematical modelling of the problem is done via nanofluid effective correlations comprising the influences of freezing temperature, molecular diameter and similarity transformations. The results for multiple parameters are plotted and provide comprehensive discussion. From the analysis, it is examined that Al2O3-H2O nanofluid motion drops by strengthening Lorentz forces. The temperature in the nanofluid (Al2O3-H2O) is improved by inducing viscous dissipation effects (Ec number), surface convection (Biot number) and thermal radiations (Rd). Moreover, the shear stresses at the surface decreased due to higher magnetic field effects and rises due to velocity slip. A significant rise in Local Nusselt number is observed due to thermal radiations and Biot effects. Finally, enhanced heat transport mechanism in Al2O3-H2O is examined than a conventional liquid. Therefore, nanofluids are better for industrial applications and the uses of conventional liquids are limited due to low thermal conductivity.

Abstract The investigation of thermal performance in the nanofluids for unsteady boundary layer flow by considering the impacts of suction/injection is a significant research area in the field of fluid dynamics. This flow situation is broadly used in aerodynamics and space sciences as well. The model is formulated for ND-H2O and Ag-H2O nanofluids and then tackled numerically and captured the dynamics of the nanofluids under the multiple flow parameters. From the results, it is investigated that both ND-H2O and Ag-H2O have high thermal performance characteristics. However, higher heat transfer performance is observed for Ag based nanofluid. Further, a graphical and tabular comparison under certain assumptions is provided to authenticate the analysis.

Aim of this article is to investigate soliton solutions of recently developed 3+1-dimensional Boiti–Leon–Manna–Pempinelli equation by utilizing newly derived approach namely, improved tanh(ϕ2)-expansion method. As a result, we succeed to secure various types of new solutions for this model including kink, periodic rational solutions. Some of the derived solutions has been discussed in the form of 2-,3-dimensional graphs and their contour plots to visualize the wave dynamics graphically. The results generated by this technique proves that it is a straightforward, robust, and effective method to generate variety of solutions and can be applied on different nonlinear models.

Abstract Now a day’s variety of nanomaterials is available, among these Aluminum Alloys AA7072 and AA705 are significant due to their thermal, physical and mechanical characteristics. These extensively used in manufacturing of spacecraft, aircraft parts and building testing. Keeping in view the significance of nanoliquids, the analysis of methanol suspended by AA7072 and AA7075 alloys under the multiple physical flow conditions is reported. The model is successfully treated by coupling of RK and shooting algorithm and examined the results for the flow regimes by altering the ingrained physical parameters. Then physical interpretation of the results discussed comprehensively. To validate the analysis, a comparison between the presented and existing is reported under certain assumptions on the flow parameters. It is found that the results are reliable inline with existing once.

All over the world, people are facing the challenge of HIV as it affects the person suffering from it, the communities and the economy of society also. A lot of work has been done on different aspects of this dangerous disease by many researchers. Our task in this work is to lighten the ignorant aspect of this disease and to provide a continuous future prediction for the spread of this virus on a larger domain. For this purpose, a modification has been done in the mathematical model of HIV CD4+T-cells. And a semi-analytical technique has been applied on the system of nonlinear differential equations, governing the human immunodeficiency virus i-e HIV infection, of CD4+T-cells, to get convergent and continuous solutions. Graphical results and residual error for the solution of the nonlinear system of differential equations are also given.

The use of improved generalized Riccati equation mapping method has been demonstrated to find some new exact travelling wave solutions to space-time fractional non-liner double dispersive equation (DDE). The equation is used for modelling wave propagation in elastic inhomogeneous Murnaghan's rod. We have used Caputo's fractional derivative to achieve the fractional version of Murnaghan's rod equation. Improved generalized Riccati equation mapping method proves to be very effective tool to find a variety of soliton solutions. As a result, we found many new and general solutions including dark, combined dark-bright, singular periodic wave, combined singular periodic wave solutions and rational solutions. We have simulated the solitons, to check their types, with the help of graphs and all the solutions obtained in this article have been verified by back substitution in original equation by using Maple 17.

An SIR-type (Susceptible-Infected-Recovered) model for the study of the spread of Ebola Virus Disease (EVD) is developed, by using conformable derivatives. Every possible way of transmission of the disease is incorporated (direct or indirect), such as funeral practices, consumption of contaminated bush meat and the environmental contamination etc. We have added an extremely important term to the model which have very high physical significance i.e., the possibility of the birth of an infected individual and the migration of an infected individual to the existing population. Well-posedness of the proposed problem has been shown by using a well-known theorem. The situations for the disease to be died out or sustain, have been discussed in the details. We found that the only disease-free situation is, the absence of flow of Ebola virus disease from the environment. We also have observed that by adopting a few strategies, such as isolation of infected individuals and careful burial of deceased bodies, the spread of EVD can be controlled. Memory effects for each case (disease-free and endemic states) are discovered (by using Khalil’s conformable transform) and plotted to make future predictions more accurately. Graphs are clearly elaborating that the problem is stable for both the equilibria states i.e., endemic state and disease-free state.

The heat transport in the nanofluids attained much interest of the researchers and engineers due to broad uses in medical sciences, paint industries, aerodynamics, wheel alignment and manufacturing of aircraft parts. Therefore, keeping in mind the paramount significance of the heat transfer, the study of Cu-nanomaterials based nanofluid is conducted. The governing nanofluid model transformed in dimensionless version via similarity transformations. For numerical simulation of the dynamics of Cu-H 2 O, RK technique with shooting algorithm is employed and presented behavior of the fluid motion, temperature, wall shear stresses and local thermal performance rate via graphical aid. It is noted that the heat transfer augmented promptly by increasing λ and volumetric fraction of Cu nanomaterial. Further, graphical and tabular comparison is also provided under certain assumptions which authenticate the study.

In this paper Chebyshev Wavelets Method (CWM) is applied to obtain the numerical solutions of fractional fourth, sixth and eighth order linear and nonlinear boundary value problems. The solutions of the fractional order problems are shown to be convergent to the integer order solution of that problem. The computational work is done successfully with the help of the proposed algorithm and hence this algorithm can be extended to other physical problems. High level of accuracy is obtained by the present method.

Abstract New exact solutions of the space–time conformable Caudrey–Dodd–Gibbon (CDG) equation have been derived by implementing the conformable derivative. The generalized Riccati equation mapping method is applied to figure out twenty-seven forms of exact solutions, which are soliton, rational, and periodic ones. Also, for some suitable values of parameters, the exact solutions are found, namely dark, bell type, periodic, soliton, singular soliton, and several others, by using the conformable derivative. These types of solutions have not been proclaimed so far. 2D and 3D graphical patterns of some solutions are also given for clarification of physical features. The conformable derivative is one of the excellent choices to solve the nonlinear conformable problems arising in theory of solitons and many other areas. The results are new and very interesting for the large community of researchers working in the field of mathematics and mathematical physics.

Currently, thermal improvement in the nanofluids over a curved Riga sheet is a topic of interest and attained popularity among the researchers. Therefore, the colloidal suspension of water suspended by A l 2 O 3 and γ A l 2 O 3 over a curved Riga surface is modeled for the heat transfer analysis. The nondimensionalization of the model is accomplished via invertible variables. On the basis of dynamic viscosities and thermal conductivities of A l 2 O 3 and γ A l 2 O 3 nanoparticles, two nanofluid models developed over a semi-infinite region. Then, the models solved numerically and found graphical results for the flow characteristics, thermophysical properties and local thermal performance rate by altering the pertinent flow parameters. It is examined that the fluid motion rapidly decreases for γ A l 2 O 3 − H 2 O and momentum boundary layer region decreases. The squeezed and curvature parameters lead to reduce in the nanofluid velocity. The temperature of more magnetized enhances significantly. Thermophysical characteristics of the nanofluids enhance for higher volumetric fraction factor. More heat transfer at the Riga surface for higher M and R .

The purpose of this work is to investigate the heat transport on water suspended by aluminium alloy nanomaterials. The analysis is conducted by incorporating the influence of imposed magnetic field and viscous dissipation over convective surface. The self-similar version of the model is treated numerically and the results for the flow field are presented. It is perceived that the velocity of AA7072-H 2 O and AA7075-H 2 O declines for stronger magnetic field effects. Due to convective condition, the temperature rises abruptly. Moreover, increasing trends in the local heat transfer rate are examined for higher Biot effects.

In this paper, we investigated the non-linear Schrödinger equation (NLS) to extract optical soliton solutions by implementing the extended Sinh–Gordon equation expansion method (ShGEEM). Optical soliton solutions included bright, dark, combined bright-dark, singular soliton combined singular soliton solutions, and singular periodic wave solutions. Our new results have been compared to these in the literature. Also, graphical analysis was presented with 3D and contour graphs to understand the physics of obtained solutions.

Hidden effects of MHD and thermal radiations for a viscous dissipative nanofluids (Al2O3 − H2O and γAl2O3 − H2O) are taken under consideration. The models are formulated by implementing the suitable similarity transformations. Then, two models are discussed mathematically by using RK scheme together with shooting method. The results for flow regimes, coefficient of skin friction, thermophysical characteristics, and heat transfer coefficient are pictured and discussed comprehensively by changing the pertinent flow parameters. It is observed that the nanofluids velocity increases abruptly for higher Hartree pressure gradient. For assisting flow situation, the velocity F′(η) increases and abrupt decreasing behavior is examined for opposing flow case. The composition of Al2O3 − H2O, and γAl2O3 − H2O becomes more dense for high volume fraction therefore, drops in the velocity field is noted. The temperature of Al2O3 − H2O, and γAl2O3 − H2O rises rapidly by varying opposing flow parameter γ < 0 and high volume fraction ϕ. Also, the temperature β(η) declines abruptly for parameter λ.

Thermal radiations and effective characteristics of the nanoparticles play a vibrant role in the heat transfer enhancement. This work aims to investigate the influence of thermophysical characteristics of nano-diamond (ND) and silver (Ag) nanoparticles on the heat transfer intensifications in the nonlinear radiative and dissipative flow over an arched geometry. A self-similar coupled nature of the nanofluid model is acquired via similarity transformations. Furthermore, the mathematical treatment of the model is carried out by the RK scheme. Moreover, stimuli of various pertinent flow parameters on the velocity, heat transfer, streamlines, and isotherms are examined graphically.

In this paper, we likewise calculate exact langrage multiplier and it progressively connects to the definite results of telegraph equations. This study witnesses the advantages of Exact Lagrange multipliers and their coupling with Variational Iteration Method. Computational work reflects that use of exact Lagrange multipliers reduces the computational work to a tangible level and also increase the level of accuracy. Moreover, exact Lagrange multiplier reduce the successive application of integral operator and are more user friendly. Keywords: Exact Langrage Multiplier (ELM), Approximate Langrage Multiplier (ALM), Variational Iteration Method (VIM), Adomian’s Decomposition Method (ADM), Telegraph equations, MAPLE 13

In this paper, an unknown auxiliary parameter is inserted in Variation of Parameter Method to solve Evolution and RLW equations. The proposed algorithm is very effective and is practically well suited for use in these problems. Three practical examples are given to verify the accuracy and efficiency of the developed technique. Comparison with the standard VPM is made to show the reliability and efficiency of the proposed algorithm. Keywords: Variation of Parameter Method (VPM), Auxiliary parameter, Evolution and RLW equations

This manuscript deals with time dependent squeezed flow of magneto-nanofluid between two parallel disks. The effects of viscous dissipation are also taken into consideration. Base is suspended with nanoparticles of various shapes. Three different shaped nanoparticles are suspended in the base fluid. Hamilton and Crosser’s model for effective thermal conductivity of the nanofluid has been considered for the analysis. For dynamic viscosity, Brinkman model is used. The suitable similarity transforms are used to transform the equations governing the flow into a set of ordinary differential equations. Three techniques are used to solve the resulting system, namely Adomian’s decomposition method (ADM), Variation of parameters method (VPM) and Runge-Kutta numerical method. The comparison between solutions has been made and found an excellent agreement. The influence of varying flow parameters on velocity and temperature fields is also portrayed graphically. The numerical values for skin fraction and local heat transfer are calculated for emerging parameters. Furthermore, the values for some constants appeared in the analytical solutions are also found for different parameters. Keywords: Nanofluids, Variation of parameters method (VPM), Adomian’s decomposition method (ADM), Runge-Kutta scheme, Hamilton and Crossers model

The Chebyshev Wavelet Method (CWM) is applied to evaluate the numerical solutions of some systems of linear fractional Voltera integro differential equations (FVIDEs). The applicability and validity of the proposed method is ensured by discussing some illustrative examples. The numerical results obtained by this technique are compared with the exact solutions of the problems. The error analysis reveals that the accuracy of the present method is higher than any existing numerical method.

This work proposes two well-known methods, namely, Exponential rational function method (ERFM) and Generalized Kudryashov method (GKM) to seek new exact solutions of the van der Waals normal form for the fluidized granular matter, linked with natural phenomena and industrial applications. New soliton solutions such as kink, periodic and solitary wave solutions are established coupled with 2D and 3D graphical patterns for clarity of physical features. Our comparison reveals that the said methods excel several existing methods. The worked-out solutions show that the suggested methods are simple and reliable as compared to many other approaches which tackle nonlinear equations stemming from applied sciences. Keywords: Exact soliton solutions, Exponential rational function method (ERFM), Generalized Kudryashov method (GKM), Van der Walls normal form for the fluidized granular matter

A relatively new technique which is named as G′G2-expansion method is applied to attain exact solution of nonlinear fractional differential equations (NLFDEs). Exact solutions include hyperbolic solutions, trigonometric solutions and rational solutions. Fractional complex transform is used to convert fractional differential equations into ordinary differential equations. We apply it to the space-time fractional Zakharov Kuznetsov Benjamin Bona Mahony (ZKBBM) and fractional coupled Burgers equation. By using our proposed method some innovative solutions can be derived. Keywords: Zakharov Kuznetsov Benjamin Bona Mahony (ZKBBM) equation, Coupled Burgers equation, Exact solutions, G′G2-expansion method

We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses. Keywords: Exp-function method, New exact traveling wave solutions, Modified Riemann-Liouville derivative, Fractional complex transformation, Fractional order Boussinesq-like equations, Symbolic computation

Abstract In this paper, we investigate a fully implicit finite difference scheme for solving the time fractional advection–diffusion equation. The time fractional derivative is estimated using Caputo’s formulation, and the spatial derivatives are discretized using extended cubic B-spline functions. The convergence and stability of the fully implicit scheme are analyzed. Numerical experiments conducted indicate that the scheme is feasible and accurate.

In this paper, we use the modified variation of parameters method (MVPM), an elegant coupling of variation of parameters method (VPM) and Adomian's decomposition method (ADM), for finding the analytical solution of system of nonlinear fractional boundary value problems associated with obstacle. Caputo sense of fractional derivative is used to coup up with fractional term. The results are calculated in terms of series with easily computable components. The used technique is quite easy and convenient for such type problems because it has been previously applied over several nonlinear obstacle systems.

Investigation of the flow, heat and mass transfer of a nanofluid over a suddenly moved flat plate is presented using Buongiorno's model. This study is different from some of the previous studies as the effects of Brownian motion and thermophoresis on nanoparticles volume fraction are passively controlled on the boundary rather than actively. The partial differential equations governing the flow are reduced to a system of nonlinear ordinary differential equations. Viable similarity transforms are used for this purpose. A well-known numerical scheme called Runge-Kutta-Fehlberg method coupled with shooting procedure has been used to find the solution of resulting system of equations. Discussions on the effects of different emerging parameters is provided using graphical aid. A table is also given that provides the results of different parameters on local Nusselt and Sherwood numbers. The passive control model can be used to control the boundary layer thickness as well as the rate of mass transfer at the wall. Keywords: Nanofluid, Brownian motion, Thermophoresis, Nusselt number, Sherwood number

In this paper, we explore new applications of the extended Gâ²G-expansion method. We apply this method to the nonlinear CalogeroâBogoyavlinskiiâSchiff equation of fractional order. As results, some new exact traveling wave solutions are obtained which include solitary wave solutions. The traveling wave solutions are expressed by hyperbolic and trigonometric functions. Keywords: Fractional derivative, The extended Gâ²G-expansion method, CalogeroâBogoyavlinskiiâSchiff equation, Traveling wave solutions

Abstract In this paper, we solve two-dimensional modified anomalous fractional sub-diffusion equation using modified implicit finite difference approximation. The stability and convergence of the proposed scheme are analyzed by the Fourier series method. We show that the scheme is unconditionally stable and the approximate solution converges to the exact solution. A numerical example is given to show the application and feasibility of the proposed scheme.

In the present paper, we construct the analytical solutions of some nonlinear equations involving Jumarie's modified Riemann-Liouville derivative in mathematical physics; namely the space–time fractional Calogero-Degasperis (CD) equation and the space–time fractional potential Kadomtsev-Petviashvili (PKP) equation by using a simple method which is called the fractional sub-equation method. As a result, three types of exact analytical solutions are obtained. This method is more powerful and will be used in further works to establish more entirely new solutions for other kind of nonlinear fractional PDEs arising in mathematical physics.

In this article, the presented study is based on a modification in Gegenbauer wavelets method. The modeled problem is presented to analyze the phenomena of transfer of heat of rotating nanofluids in which the flow is produced by an exponentially stretching sheet. The purpose of this study is to examine the simultaneous effects of rotation of nanofluid and exponentially stretching on the shear stresses and heat transfer rate, cooling proficiency of water-based nanofluids containing Ag, Cu, Al 2 O 3 , TiO 2 , and CuO nanoparticles, and modification in Gegenbauer wavelets method to obtain the numerical solution of the said problem. A comparative analysis is presented among the outcomes obtained by modified Gegenbauer wavelets method, Runge–Kutta method of order-4, and already existing methods. The comparison shows that this modification is extremely efficient, and proposed technique could be extended for other physical problems.

This article addresses the phenomenon of boundary layer flow and heat transfer in respect of the motion of second grade viscous fluid over an unsteady stretchable surface. Optimal variational iteration method (OVIM) is employed to solve the governing differential equation. OVIM is the modified version of variational iteration method (VIM). The convergence of the obtained solution as well as the effect of pertinent parameters on the velocity components are discussed, tabulated, and graphed. Behaviour of coefficient of skin-friction is also tabulated and explained for different parameters. Comparison of residual errors sought via use of VIM and OVIM is also presented.

In this study, we apply Variation of Parameter Method (VPM) coupled with an auxiliary parameter to obtain the approximate solutions for the epidemic model for the evolution of smoking habit in a constant population. Convergence of the developed algorithm, namely VPM with an auxiliary parameter is studied. Furthermore, a simple way is considered for obtaining an optimal value of auxiliary parameter via minimizing the total residual error over the domain of problem. Comparison of the obtained results with standard VPM shows that an auxiliary parameter is very feasible and reliable in controlling the convergence of approximate solutions. Keywords: Smoking model, Variation of Parameter Method (VPM), Auxiliary parameter

In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO) equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering. Keywords: Khater method, Modified Riemann-Liouville derivative, Exact solutions, Fractional complex transform, Space-time fractional Sharma Tasso-Olever (STO) equation

Squeezing flow of an electrically conducting Casson fluid has been taken into account. The laws of conservations under the similarity transformation suggested by Wang (1976) have been used to extract a highly nonlinear ordinary differential equation governing the magneto hydrodynamic (MHD) flow. Resulting equation has been solved analytically by using the variation of parameters method (VPM). A RK-4 numerical solution has also been sought to examine the validity of analytical results. Both the solutions are found to be in an excellent agreement. Convergence of the solution is also discussed. Flow behavior under the modifying involved physical parameters is also discussed and explained in detail with the graphical aid. It is observed that magnetic field can be used as a control phenomenon in many flows as it normalizes the flow behavior. Also, squeeze number plays an important role in these types of problems and an increase in squeeze number increases the velocity profile.

This article investigates the problem related to Jeffery-Hamel with stretchable walls. The effects of mass and heat transfer are taken into account. Soret, Dufour and viscous dissipation effects are examined to scrutinize the behavior of concentration and temperature profiles between convergent/divergent stretchable channels. By using similarity variables, nonlinear partial differential equations governing the flow are reduced into the nondimensional coupled system of ordinary differential equations. Then, said flow model is tackled analytically and numerically over a prescribed domain. For analytical solution, we employed Homotopy Analysis method while for numerical once, Runge-Kutta scheme is used after reducing the system of ordinary differential equations into system of first order initial value problem. The effects of various dimensionless parameters that ingrained in the velocity, concentration and temperature fields are scrutinized graphically for convergent and divergent stretchable channels. Also, the values of skin friction coefficient, local Nusselt and Sherwood numbers are calculated analytically by using Homotopy Analysis method (HAM) for different physical parameters. Different values of quantities of physical interest (skin friction coefficient, local rate of heat and mass transfer) are tabulated for different ingrained parameters in the flow model. Finally, comparison between present results with already existing results in the literature has been made. Keywords: Second grade fluid, Diverging/converging channels, Analytical solution, Skin friction coefficient, Sherwood and Nusselt numbers

The modification of the expansion method, namely, the New Modified Simple Equation Method is presented and subsequently implemented on nonlinear partial differential equations. The implementation of the new modification is demonstrated by solving Landau-Ginburg-Higgs equation and Cahn-Allen equation, as a result many new and more general multiple wave solution consisting of Hyperbolic function solutions, trigonometric function solutions and rational function solutions are obtained. Also double solitary wave solutions are found by substituting the different values for some parameters. The method appears to be straightforward and powerful by means of symbolic computational system. Keywords: New Modified Simple Equation Method, Multiple wave solutions, Landau-Ginburg-Higgs equation, Cahn-Allen equation

In this study, we present a new modified convergent analytical algorithm for the solution of nonlinear boundary value problems by the Variation of Parameters Method (VPM). The method is based on embedding, auxiliary parameter and auxiliary linear differential operator, provides a computational advantage for the convergence of approximate solutions for nonlinear differential equations. Convergence of developed scheme is also shown and discussed in detail. Moreover, a convenient way is considered for choosing an optimal value of auxiliary parameter via minimizing the residual error over the domain of problem. The accuracy and efficiency of the proposed algorithm is established by implementing on some physical problems. The obtained graphical and numerical results clearly reflects the accuracy and convergence of the presented algorithm. Keywords: Variation of Parameters Method (VPM), Auxiliary parameter, Convergence analysis, Residual error, Boundary value problems

In this work, we have obtained some exact solutions to (3 + 1)-dimensional generalized KP Equation. The improved tanϕ(ξ)2-expansion method has been introduced to construct the exact solutions of nonlinear evolution equations. The obtained solutions include hyperbolic function solutions, trigonometric function solutions, exponential solutions, and rational solutions. Our study has added some new varieties of solutions to already available solutions. It is also worth mentioning that the computational work has been reduced significantly. Keywords: Improved tanϕ(ξ)2-expansion method, Hyperbolic function solution, Trigonometric function solution, Rational solution, (3 + 1)-dimensional generalized KP equation

This article deals with finding some exact solutions of nonlinear fractional differential equations (NLFDEs) by applying a relatively new method known as G′G2-expansion method. Solutions of space–time fractional Sharma-Tasso-Olever (STO) equation of fractional order and (3+1)-dimensional KdV-Zakharov Kuznetsov (KdV-ZK) equation of fractional order are reckoned to demonstrate the validity of this method. The fractional derivative version of modified Riemann–Liouville, linked with Fractional complex transform is employed to transform fractional differential equations into the corresponding ordinary differential equations. Keywords: Sharma Tasso-Olever (STO) equation, (3+1)-Dimensional KdV-Zakharov Kuznetsov (KdV-ZK) equation, Exact solutions, G′G2-expansion method

The aim of this letter is to investigate the unsteady flow of incompressible fluid in the presence of thermal and concentration gradients over a bi-directional stretchable flat sheet that placed in nonporous media. The nondimensional form of the model is successfully obtained after utilizing the set of similarity transformations and necessary mathematical calculations. Then, resulting two-point boundary value problem is sort out numerically with the help of Runge-Kutta scheme after reducing the flow model into a set of initial value problem. Influence of the various flow parameters on the velocity, thermal and concentration fields discussed comprehensively by means of graphs. Furthermore, quantities of practical and physical interest such as skin friction coefficient, local rate of heat and mass transfer studied with the help of graphical aid. Finally, major outcomes of the study highlighted in the last section of the manuscript. Keywords: Stretching surface, Cartesian coordinates, Soret effects, Dufour effects, Skin friction coefficient, Local Nusselt number, Local Sherwood number, Runge-Kutta scheme

In this work, we established some exact particular solutions with parameters for Modified KdV-ZK Equation. The improved tanϕ(ξ)2-expansion method is introduced to construct exact particular solutions of nonlinear evolution equations. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. The method appears to be effective mathematical tool for solving nonlinear evolution equations (NLEEs) in applied mathematics and engineering.

Heat transfer analysis for the squeezing flow of a Casson fluid between parallel circular plates has been presented. Viable mathematical model has been constructed by using conservation laws coupled with suitable similarity transforms. This model ends up on a set of two highly nonlinear ordinary differential equations. Resulting equations have been solved by using a well-known analytical technique homotopy perturbation method (HPM). A numerical solution using forth order Runge–Kutta method has also been sought to support our analytical solution and the comparison shows an excellent agreement. Flow behavior under altering involved physical parameters is also discussed and explained in detail with graphical aid. For the presented problem, values of parameters are restricted. Analysis is carried out using the following ranges of parameters; squeeze number (-4⩽S⩽4), Casson fluid parameter (0.1⩽β⩽∞), Prandtl number (0.1⩽Pr⩽0.7), Eckert number (0.1⩽Ec⩽0.7) and 0.1⩽δ⩽0.4. Increase in velocity for squeeze number and Casson fluid parameter is observed. Temperature profile is found to be decreasing function of squeeze number and Casson fluid parameter and increasing function of Pr, Ec and δ.

Currently, thermal investigation in hybrid colloidal liquids is noteworthy. It has applications in medical sciences, drug delivery, computer chips, electronics, the paint industry, mechanical engineering and to perceive the cancer cell in human body and many more. Therefore, the study is carried out for 3D magnetized hybrid nanofluid by plugging the novel Cattaneo–Christov model and thermal radiations. The dimensionless version of the model is successfully handled via an analytical technique. From the reported analysis, it is examined that Graphene Oxide-molybdenum disulfide/C2H6O2-H2O has better heat transport characteristics and is therefore reliable for industrial and technological purposes. The temperature of Graphene Oxide GO-molybdenum disulfide/C2H6O2-H2O enhances in the presence of thermal relaxation parameter and radiative effects. Also, it is noted that rotational velocity of the hybrid nanofluid rises for stronger magnetic parameter effects. Moreover, prevailed behavior of thermal conductivity of GO-molybdenum disulfide/C2H6O2-H2O is detected which shows that hybrid nanofluids are a better conductor as compared to that of a regular nanofluid.

The flow of nanofluid over a curved Riga surface is a topic of interest in the field of fluid dynamics. A literature survey revealed that the impacts of freezing temperature and the diameter of nanoparticles on the heat transfer over a curved Riga surface have not been examined so far. Therefore, the flow of nanoparticles, which comprises the influences of freezing temperature and nanoparticle diameter in the energy equation, was modeled over a curved Riga surface. The model was reduced successfully in the nondimensional version by implementing the feasible similarity transformations and effective models of nanofluids. The coupled nonlinear model was then examined numerically and highlighted the impacts of various flow quantities in the flow regimes and heat transfer, with graphical aid. It was examined that nanofluid velocity dropped by increasing the flow parameters γ and S, and an abrupt decrement occurred at the surface of the Riga sheet. The boundary layer region enhances for larger γ. The temperature distribution was enhanced for a more magnetized nanofluid, and the thermal boundary layer increased with a larger R parameter. The volume fraction of the nanoparticles favors the effective density and dynamic viscosity of the nanofluids. A maximum amount of heat transfer at the surface was observed for a more magnetized nanofluid.

Thermal transport analysis in colloidal suspension is significant from industrial, engineering, and technological points of view. It has numerous applications comprised in medical sciences, chemical and mechanical engineering, electronics, home appliances, biotechnology, computer chips, detection of cancer cells, microbiology, and chemistry. The carbon nanomaterials have significant thermophysical characteristics that are important for thermal transport. Therefore, the thermal transport in H2O composed by single and multiwalled carbon nanotubes is examined. The length and radius of the nanomaterials is in range of 3 μm ≤ L* ≤ 70 μm and 10 nm ≤ d ≤ 40 nm, respectively. The problem is modelled over a curved stretching geometry by inducing the velocity slip and thermal jump conditions. The coupling of Runge-Kutta (RK) and shooting technique is adopted for the solution. From the analysis it is perceived that the heat transfer at the surface drops for stretching. The heat transfer rate prevailed for Single walled carbon nanotubes SWCNTs-H2O colloidal suspension. The suction and stretching of the surface resist the shear stresses and more shear stress trends are investigated for larger curvature.

In the present study, our aim is to present a novel model for the flow of hybrid nanofluids in oblique channels. Copper and aluminum oxide have been used to obtain a novel hybrid nanofluid. The equations that govern the flow of hybrid nanofluids have been transformed to a set of nonlinear equations with the implementation of self-similar variables. The resulting system is treated numerically by using coupled shooting and Runge–Kutta (R-K) scheme. The behavior of velocity and temperature is examined by altering the flow parameters. The cases for narrowing (convergent) and opening (divergent) channels are discussed, and the influence of various parameters on Nusselt number is also presented. To indicate the reliability of the study, a comparison is made that confirms the accuracy of the study presented.

The flow of nanofluid between infinite parallel plates suspended by micro-cantilever sensors is significant. The analysis of such flows is a rich research area due to the variety of applications it has in chemical, biological and medical sciences. Micro-cantilever sensors play a significant role in accurately sensing different diseases, and they can be used to detect many hazardous and bio-warfare agents. Therefore, flow water and ethylene glycol (EG) composed by γ-nanoparticles is used. Firstly, the governing nanofluid model is transformed into two self-similar nanofluid models on the basis of their effective models. Then, a numerical method is adopted for solution purposes, and both the nanofluid models are solved. To enhance the heat transfer characteristics of the models, the effective Prandtl model is ingrained in the energy equation. The velocity F’(η) decreases with respect to the suction of the fluid, because more fluid particles drags on the surface for suction, leading to an abrupt decrement in F’(η). The velocity F’(η) increases for injection of the fluid from the upper end, and therefore the momentum boundary layer region is prolonged. A high volume fraction factor is responsible for the denser characteristics of the nanofluids, due to which the fluids become more viscous, and the velocity F’(η) drops abruptly, with the magnetic parameters favoring velocity F’(η). An increase in temperature β ( η ) of Al2O3-H2O and γAl2O3-C2H6O2 nanofluids was reported at higher fraction factors with permeable parameter effects. Finally, a comparative analysis is presented by restricting the flow parameters, which shows the reliability of the study.

Thermal enhancement and irreversible phenomena in colloidal suspension (Al2O3-H2O) is a potential topic of interest from the aspects of industrial, mechanical and thermal engineering; heat exchangers; coolant car radiators; and bio-medical, chemical and civil engineering. In the light of these applications, a colloidal analysis of Al2O3-H2O was made. Therefore, a colloidal model is considered and treated numerically. The significant influences of multiple parameters on thermal enhancement, entropy generation and Bejan parameter are examined. From the presented colloidal model, it is explored that Al2O3-H2O is better for the applications of mechanical and applied thermal engineering. Moreover, fraction factor tiny particles are significant parameters which enhanced the thermal capability of the Al2O3-H2O suspension.

In this paper, exp-function method is used to construct generalized solitary solutions of the Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation. It is shown that the exp-function method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool to solve such nonlinear equations.The performance of the method is reliable, efficient and it gives useful exact solutions.

This article is dedicated to analyzing the heat transfer in the flow of water-based nanofluids in a channel with non-parallel stretchable walls. The magnetohydrodynamic (MHD) nature of the flow is considered. Equations governing the flow are transformed into a system of nonlinear ordinary differential equations. The said system is solved by employing two different techniques, the variational iteration method (VIM) and the Runge-Kutta-Fehlberg method (RKF). The influence of the emerging parameters on the velocity and temperature profiles is highlighted with the help of graphs coupled with comprehensive discussions. A comparison with the already existing solutions is also made, which are the special cases of the current problem. It is observed that the temperature profile decreases with an increase in the nanoparticle volume fraction. Furthermore, a magnetic field can be used to control the possible separation caused by the backflows in the case of diverging channels. The effects of parameters on the skin friction coefficient and Nusselt number are also presented using graphical aid. The nanoparticle volume fraction helps to reduce the temperature of the channel and to enhance the rate of heat transfer at the wall.