• The nonlinear KdV equation is a functional description for modeling ion-acoustic waves in plasma and long internal waves in a density-stratified ocean. • This paper proposes the q-HASTM for generalized fuzzy fractional KdV equation. • The fuzzy velocity profiles at different spatial positions with crisp and fuzzy conditions are investigated. • The obtained results are compared with existing works to confirm effectiveness of the method. The nonlinear Kortewege-de Varies (KdV) equation is a functional description for modelling ion-acoustic waves in plasma, long internal waves in a density-stratified ocean, shallow-water waves and acoustic waves on a crystal lattice. This paper focuses on developing and analysing a resilient double parametric analytical approach for the nonlinear fuzzy fractional KdV equation (FFKdVE) under gH-differentiability of Caputo fractional order, namely the q -Homotopy analysis method with the Shehu transform ( q -HASTM). A triangular fuzzy number describes the Caputo fractional derivative of order α , 0 < α ≤ 1 , for modelling problem. The fuzzy velocity profiles with crisp and fuzzy conditions at different spatial positions are investigated using a robust double parametric form-based q -HASTM with its convergence analysis. The obtained results are compared with existing works in the literature to confirm the efficacy and effectiveness of the method.

• Memory response in a nonlocal micropolar double porous material with voids is analyzed. • Moore-Gibson-Thompson equation with variable thermal conductivity is introduced. • The normal mode technique is employed to obtain the field variables analytically. • Effects of various key parameters are observed graphically. The present study enlightens the two-dimensional analysis of the thermo-mechanical response for a micropolar double porous thermoelastic material with voids (MDPTMWV) by virtue of Eringen’s theory of nonlocal elasticity. Moore-Gibson-Thompson (MGT) heat equation is introduced to the considered model in the context of memory-dependent derivative and variable conductivity. By employing the normal mode technique, the non-dimensional coupled governing equations of motion are solved to determine the analytical expressions of the displacements, temperature, void volume fractions, microrotation vector, force stress tensors, and equilibrated stress vectors. Several two-dimensional graphs are presented to demonstrate the influence of various parameters, such as kernel functions, thermal conductivity, and nonlocality. Furthermore, different generalized thermoelasticity theories with variable conductivity are compared to visualize the variations in the distributions associated with the prior mentioned variables. Some particular cases are also discussed in the presence and absence of different parameters.

• Fractional Reduced Differential Method (FRDTM) is established and applied to find the closed-form solution of time-fractional diffusion equation and Allen-Cahn equation arising in oil pollution. • The obtained results using FRDTM has been compared with the exact solution, MVIA-I, MVIA-II, MQM, LLWM, and ADM for integer order. • The simulation results indicate an excellent accordance with the exact solution as compared to any other available method in the literature. • FRDTM gives fast convergence and provides highly accurate numerical results. • The main advantage of this method is its implementation on time-fractional order nonlinear PDEs without discretization and linearization. In this article, non-linear time-fractional diffusion equations are considered to describe oil pollution in the water. The latest technique, fractional reduced differential transform method (FRDTM), is used to acquire approximate solutions of the time fractional-order diffusion equation and two cases of Allen–Cahn equations. The acquired results are collated with the exact solutions and other results from literature for integer-order α , which reveal that the proposed method is effective. Hence, FRDTM can be employed to obtain solutions for different types of nonlinear fractional-order IVPs arising in engineering and science.

• Analytical solutions of (2+1)-coupled dispersive long wave equations (DLWEs) are obtainedby using Similarity transformations method via Lie group. • Some of the results derived in the previous findings are deduced from these results. • DLWEs can be observed in an open sea or in wide channels. • The analytical solutions show elastic multisolitons, single soliton, doubly solitons, stationary, kink and parabolic profiles. This work is devoted to get a new family of analytical solutions of the (2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth, and can be observed in an open sea or in wide channels. The solutions are obtained by using the invariance property of the similarity transformations method via one-parameter Lie group theory. The repeated use of the similarity transformations method can transform the system of PDEs into system of ODEs. Under adequate restrictions, the reduced system of ODEs is solved. Numerical simulation is performed to describe the solutions in a physically meaningful way. The profiles of the solutions are simulated by taking an appropriate choice of functions and constants involved therein. In each animation, a frame for dominated behavior is captured. They exhibit elastic multisolitons, single soliton, doubly solitons, stationary, kink and parabolic nature. The results are significant since these have confirmed some of the established results of S. Kumar et al. (2020) and K. Sharma et al. (2020). Some of their solutions can be deduced from the results derived in this work. Other results in the existing literature are different from those in this work.

• The optimal system of Lie subalgebra is employed to analyze weakly coupled B-type Kadomtsev-Petviashvili equations. • We started with Lie infinitesimals, potential vector fields, and a one-dimensional optimal system. • Symmetry reductions are applied to obtain various explicit solutions. • Various wave profiles are used to demonstrate the dynamics of several closed-form solutions. • In the field of advanced research and development, such investigations are fully supported. In the case of negligible viscosity and surface tension, the B-KP equation shows the evolution of quasi-one-dimensional shallow-water waves, and it is growingly used in ocean physics, marine engineering, plasma physics, optical fibers, surface and internal oceanic waves, Bose-Einstein condensation, ferromagnetics, and string theory. Due to their importance and applications, many features and characteristics have been investigated. In this work, we attempt to perform Lie symmetry reductions and closed-form solutions to the weakly coupled B-Type Kadomtsev-Petviashvili equation using the Lie classical method. First, an optimal system based on one-dimensional subalgebras is constructed, and then all possible geometric vector yields are achieved. We can reduce system order by employing the one-dimensional optimal system. Furthermore, similarity reductions and exact solutions of the reduced equations, which include arbitrary independent functional parameters, have been derived. These newly established solutions can enhance our understanding of different nonlinear wave phenomena and dynamics. Several three-dimensional and two-dimensional graphical representations are used to determine the visual impact of the produced solutions with determined parameters to demonstrate their dynamical wave profiles for various examples of Lie symmetries. Various new solitary waves, kink waves, multiple solitons, stripe soliton, and singular waveforms, as well as their propagation, have been demonstrated for the weakly coupled B-Type Kadomtsev-Petviashvili equation. Lie classical method is thus a powerful, robust, and fundamental scientific tool for dealing with NPDEs. Computational simulations are also used to prove the effectiveness of the proposed approach.

In the fields of oceanography, hydrodynamics, and marine engineering, many mathematicians and physicists are interested in Burgers-type equations to show the different dynamics of nonlinear wave phenomena, one of which is a (3+1)-dimensional Burgers system that is currently being studied. In this paper, we apply two different analytical methods, namely the generalized Kudryashov (GK) method, and the generalized exponential rational function method, to derive abundant novel analytic exact solitary wave solutions, including multi-wave solitons, multi-wave peakon solitons, kink-wave profiles, stripe solitons, wave-wave interaction profiles, and periodic oscillating wave profiles for a (3+1)-dimensional Burgers system with the assistance of symbolic computation. By employing the generalized Kudryashov method, we obtain some new families of exact solitary wave solutions for the Burgers system. Further, we applied the generalized exponential rational function method to obtain a large number of soliton solutions in the forms of trigonometric and hyperbolic function solutions, exponential rational function solutions, periodic breather-wave soliton solutions, dark and bright solitons, singular periodic oscillating wave soliton solutions, and complex multi-wave solutions under various family cases. Based on soft computing via Wolfram Mathematica, all the newly established solutions are verified by back substituting them into the considered Burgers system. Eventually, the dynamical behaviors of some established results are exhibited graphically through three - and two-dimensional wave profiles via numerical simulation.

• The generalized (3+1)-dimensional nonlinear wave equation with gas bubbles in fluids is studied. • The generalized nonlinear equation provides a series of closed-form wave solutions that exhibit various dynamical wave forms. • The generalized exponential rational function approach was applied to obtain closed-form wave solutions. • We produce newly formed exact solitary wave profiles. • Graphical representations of achieved soliton solutions depict a variety of specific wave profiles. Nonlinear evolution equations (NLEEs) are frequently employed to determine the fundamental principles of natural phenomena. Nonlinear equations are studied extensively in nonlinear sciences, ocean physics, fluid dynamics, plasma physics, scientific applications, and marine engineering. The generalized exponential rational function (GERF) technique is used in this article to seek several closed-form wave solutions and the evolving dynamics of different wave profiles to the generalized nonlinear wave equation in (3+1) dimensions, which explains several more nonlinear phenomena in liquids, including gas bubbles. A large number of closed-form wave solutions are generated, including trigonometric function solutions, hyperbolic trigonometric function solutions, and exponential rational functional solutions. In the dynamics of distinct solitary waves, a variety of soliton solutions are obtained, including single soliton, multi-wave structure soliton, kink-type soliton, combo singular soliton, and singularity-form wave profiles. These determined solutions have never previously been published. The dynamical wave structures of some analytical solutions are graphically demonstrated using three-dimensional graphics by providing suitable values to free parameters. This technique can also be used to obtain the soliton solutions of other well-known equations in engineering physics, fluid dynamics, and other fields of nonlinear sciences.

• Mathematical model of tsunami waves has been analysed by fractional approach. • This model is governed by coupled system of non-linear partial differential equations. and results are obtained by using fractional reduced differential transform method (FRDTM). • The coastal slope and sea depth effects on tsunami wave velocity and run-up height has been demonstrated. • Advantage of FRDTM is that it has no specific requirements for nonlinear operators, discretization, linearization, transformation, or perturbation. • One of the distinguishing practical features of this technique is its implementation on time-fractional order nonlinear PDEs without using discretization and linearization. This paper presents a fractional approach to study the mathematical model of tsunami wave propagation along a coastline of an ocean. Fractional Reduced Differential Transform Method (FRDTM) is used to analyze this model. The present model has been studied on the shallow-water assumption. It is represented by a time-fractional coupled system of non-linear partial differential equations. Solutions to the proposed model for different coastal slopes and ocean depths have been obtained. Effects of coast slope and sea depth variations on tsunami wave velocity and wave amplification have been demonstrated at different time levels and different orders α . The obtained results are compared with Elzaki Adomian Decomposition Method (EADM) to validate the proposed method for an order α = 1 .

• A Boussinesq equation is investigated using the Lie symmetry approach. • By applying Lie symmetry analysis, we obtain some novel exact solutions. • It can be used to solve a wide range of nonlinear PDEs in applied mathematics and marine engineering. • Elucidate the obtained exact solutions by employing 3D and 2D graphics of different types of wave profiles. • Various multiple solitons and different wave profiles are demonstrated for the Boussinesq equation. This paper systematically investigates the exact solutions to an extended (2+1)-dimensional Boussinesq equation, which arises in several physical applications, including the propagation of shallow-water waves, with the help of the Lie symmetry analysis method. We acquired the vector fields, commutation relations, optimal systems, two stages of reductions, and exact solutions to the given equation by taking advantage of the Lie group method. The method plays a crucial role to reduce the number of independent variables by one in each stage and finally forms an ODE which is solved by taking relevant suppositions and choosing the arbitrary constants that appear therein. Furthermore, Lie symmetry analysis (LSA) is implemented for perceiving the symmetries of the Boussinesq equation and then culminating the solitary wave solutions. The behavior of the obtained results for multiple cases of symmetries is obtained in the present framework and demonstrated through three-and two-dimensional dynamical wave profiles. These solutions show single soliton, multiple solitons, elastic behavior of combo soliton profiles, and stationary waves, as can be seen from the graphics. The outcomes of the present investigation manifest that the considered scheme is systematic and significant to solve nonlinear evolution equations.

• time-fractional Ito equation has been solved utilizing the RBF-FD method. • meshfree method based on MQ-RBF for spatial discretization are considered. • spatial derivatives are discretized via the multiquadric RBF. • some useful theorems are established for stability and convergence. • results are compared with Kudryashov and tanh method solutions. In this manuscript, a meshfree numerical scheme based on radial basis function-finite difference (RBF-FD) method is proposed to solve the time-fractional Ito equation. First, the temporal derivative is approximated through the finite difference technique. Then, the spatial derivatives are discretized via the multiquadric RBF. Furthermore, some useful theorems are discussed to establish the stability and convergence of the proposed numerical scheme. Finally, some test problems are presented and the results are compared with the acquired Kudryashov method and tanh method solutions to establish the computational efficiency and applicability of the proposed method.

• A generalized coupled cubic Schrödinger–Korteweg-de Vries equations is studied. • This coupled model simulates the interaction of long and short waves which is important in many domains of applied sciences and engineering. • The modified Sardar sub-equation is used. • New solitons and solitary wave solutions are achieved. • Some plots are presented to clearly illustrate the dynamic features of these solutions. The Kortewegde Vries (KdV) equation represents the propagation of long waves in dispersive media, whereas the cubic nonlinear Schrödinger (CNLS) equation depicts the dynamics of narrow-bandwidth wave packets consisting of short dispersive waves. A model that couples these two equations seems intriguing for simulating the interaction of long and short waves, which is important in many domains of applied sciences and engineering, and such a system has been investigated in recent decades. This work uses a modified Sardar sub-equation procedure to secure the soliton-type solutions of the generalized cubic nonlinear Schrödinger–Korteweg-de Vries system of equations. For various selections of arbitrary parameters in these solutions, the dynamic properties of some acquired solutions are represented graphically and analyzed. In particular, the dynamics of the bright solitons, dark solitons, mixed bright-dark solitons, W-shaped solitons, M-shaped solitons, periodic waves, and other soliton-type solutions. Our results demonstrated that the proposed technique is highly efficient and effective for the aforementioned problems, as well as other nonlinear problems that may arise in the fields of mathematical physics and engineering.

• The proposed technique is very simple, fast, easy to implement but very efficient. • The technique solves the nonlinear partial differential equations directly without transforming them into heat equation or ordinary differential equations. • The technique does not require unnecessary integration or calculation of weight functions as the case of other numerical methods for instance Galerkin methods, spectral methods etc. • The technique does not require much computation effort. Due to the use of Hermite splines, the easily solvable banded system of algebraic equations is obtained. • The efficiency and robustness of the technique are shown by numerically solving five examples of these three non-linear partial differential equations for different parameters. In this paper, a robust Hermite collocation technique is proposed to find the numerical solution of generalized Burgers-Huxley and Burgers-Fisher equations as well as modified Burgers’ equation. In this technique, Hermite collocation method with fifth order Hermite splines have been used to approximate the solution variable and its spatial derivatives. Crank-Nicolson finite difference scheme is applied on time derivatives. The quasilinearization technique is used to linearize the nonlinear terms in the equation. Von-Neumann method is applied to show stability of the proposed technique. Robustness of proposed technique is shown by solving five test examples of these three equations with different parameters. The computed numerical results are better than the results from other techniques compared in this paper and are also matched well with the exact solutions.

• The generalized unstable nonlinear Schrodinger equation is studied. • The modified Sardar sub-equation and q-homotopy analysis transform methods are used. • Some new soliton and numerical solutions have been established. • The obtained results are presented by Figures. In this paper, a more general form of unstable nonlinear Schrödinger equation which describe the time evolution of disturbances in marginally stable or unstable media is studied. A new modification of the Sardar sub-equation method is discussed and employed to retrieve solitons and other solutions of the suggested nonlinear model. A variety of solutions, including bright solitons, dark solitons, singular solitons, combo bright-singular solitons, periodic, exponential, and rational solutions are provided with considerable physical perspective. Using the q-homotopy analysis algorithm in combination with the Laplace transform, we present the approximate solutions of the bright and dark solitons, including the physical nature of the attained solutions. The computation complexity and results indicate that the given techniques are simple, effective, uncomplicated, and that they may be used to a wide range of unstable and stable nonlinear evolution equations encountered in mathematics, mathematical physics, and other applied disciplines.

The current paper explores the behavior of the thermal radiation on the time-independent flow of micropolar fluid past a vertical stretching surface with the interaction of a transverse magnetic field. The effect of thermo-diffusion (Soret) along with the heat source is incorporated to enhance the thermal properties. Also, the convective solutal condition is considered that affects the mass transfer phenomenon. The transformed equations are modeled using suitable similarity transformation. However, the complex coupled equations are handled mathematically employing the Runge-Kutta-Felhberg method. The behavior of characterizing parameters on the flow phenomena as well as the engineering coefficients are displayed via graphs and the validation of the current outcome is reported with the previously published results in particular cases.

The time-fractional Klein-Gordon equation was analyzed in this article using the natural transform decomposition method (NTDM) and the iterative Shehu transform method (ISTM) with a singular kernel derivative. We employed natural transform (NT) and Shehu transform (ST), followed by inverse natural transform and inverse Shehu transform respectively, to obtain the solution. The numerical simulations are provided to ensure that the methods under consideration are efficient. The current framework captures the behaviour of the obtained findings for various fractional orders. The findings of this study indicate that the proposed methods are effective and reliable in analyzing fractional differential equations.

• Vortex shedding dynamics for flow past an isolated circular cylinder subjected to an oscillatory upstream is numerically investigated. • Control of wake vortices is achieved with the aid of two small rotors located in the aft of the main cylinder. • Temporal Streamline and Streakline flow visuals delineate effective control of wake vortices, under the influence of harmonic forcing. • The efficacy of the flow control is tested further, when the circular cylinder was flexibly mounted. Understanding and control of wake vortices past a circular cylinder is a cardinal problem of interest to ocean engineering. The wake formation and vortex shedding behind a variety of ocean structures such as spars, are subjected to fatigue failure limiting their life span. The additional influences due to ocean waves and currents further exacerbate these effects. In the present study, flow past an isolated circular cylindrical structure subjected to an oscillatory upstream are numerically investigated. These studies involve high resolution simulations over the low Reynolds number range (100–200). Although the practical range of interest is in high Reynolds number range of 10 3 - 10 5 , the flow physics and a number of qualitative and quantitative aspects are similar to the low Reynolds number flows. In the high Reynolds number range, statistical averaging tools in conjunction with suitable closure models would be necessary. The control of wake vortices is achieved with the aid of two small rotors located in the aft of the main cylinder. A control algorithm was coupled to determine the quantum of actuation to the rotating elements. Although control of wake vortices was observed for harmonic in-let forcing, residual vortical structures were found to persist at higher amplitudes of oscillation. To study the efficacy of this control, numerical simulations were further extended, when the circular cylinder was flexibly mounted. The control of flow induced vibrations was observed to be reasonably effective in controlling the wake generated behind the main cylinder due to oscillatory upstream.

• In case of shell instability failure mode (SIFM) as the number of stiffener increases critical buckling pressure increases, where as in case of general instability failure mode (GIFM) as the stiffeners increases the critical buckling pressure decreases. • The more (affine) failure mode out of SIFM and GIFM can be determined by the lower buckling pressure out of two buckling pressure obtained from numerical analysis of FE models of both failure modes. • For shell instability failure mode (SIFM), FE analysis and equation given by Greiner closely matches with the experimental results. • In case of models with 9 ring stiffeners (model 7 & 8) there is an equal possibility of having both failure modes namely SIFM and GIFM. Submarine pressure hulls, fire-tube boilers, vacuum tanks, oil well casings, submersibles, underground pipelines, tunnels, rocket motor casing, etc., are some of the examples of thin cylindrical shell structures which collapse due to buckling under uniform pressure. To enhance the buckling strength of bare cylindrical shells, one of the best solutions is to stiffen them with ring stiffeners. In this work in order to predict the shell instability failure mode (SIFM) and general instability failure mode (GIFM) FE models are generated and analysed using buckling analysis of general-purpose FE software ANSYS. The numerical results obtained using FE analysis are compared with published analytical and experimental results. Hence in the present study efforts are taken to develop FE models to predict global and shell instability failure modes of externally ring stiffened cylindrical shells by using linear FE analysis. It is proposed to use full/half bare cylindrical shell FE models (L/R ratio upto 200) to determine SIFM and FE models with shell281- Beam189 (for stiffeners) can be used to determine GIFM. The developed FE models are validated by comparing numerical results with experimental results published by Seleim and Roorda [25] . By using both proposed FE models it is possible to predict the failure modes namely SIFM and GIFM, comparing their values of critical buckling pressures. The lower pressure value can indicate the possible failure mode.

The research article is the analysis of wave propagation in an initially stressed micropolar fractional-order derivative thermoelastic diffusion medium with voids. The governing equations in the context of generalized fractional-order derivative thermo-elasticity are formulated and the velocity equations are obtained. The plane wave solution of these equations indicates the existence of six plane waves, namely coupled longitudinal displacement ( c L ), coupled thermal ( c T ), coupled mass diffusion ( c M D ), coupled longitudinal void volume fraction ( c V ), coupled transverse displacement ( c T D ), and coupled transverse micro-rotational ( c T M ) waves. The sets of coupled waves ( c L ), ( c T ), ( c M D ) and ( c V ) are found to be dispersive, attenuating and influenced by the presence of thermal, diffusion and voids parameters in the medium. The speeds of coupled transverse displacement ( c T D ), and coupled transverse micro-rotational ( c T M ) waves are not affected by thermal, diffusion and void parameters. The speeds of the plane waves, c L , c T , c M D , and c V are computed for a particular material and plotted against the thermal parameter, frequency, initial stress, diffusion and void parameters.

• Solution of the mathematical model of tsunami wave by Elzaki Adomian Decomposition Method has been studied. • How tsunami wave velocity and run-up height are affected by the coast slope and sea depth are demonstrated. • It can be deduced from this work that, as the tsunami approaches the shore, its velocity reduces and the wave amplitude shoots up initially and then as the wave approaches the coastline, it decreases due to shoaling. • The result shows that the speed and height of tsunami waves are inversely proportional to the depth of the ocean. The paper deals with the study of the mathematical model of tsunami wave propagation along a coastline of an ocean. The model is based on shallow-water assumption which is represented by a system of non-linear partial differential equations. In this study, we employ the Elzaki Adomian Decomposition Method (EADM) to successfully obtain the solution for the proposed model for different coastal slopes and ocean depths. How tsunami wave velocity and run-up height are affected by the coast slope and sea depth are demonstrated. The Adomian Decomposition Method together with Elzaki transform allows for solutions, without the need of any linearization or perturbation, in the form of rapidly converging series. The obtained numerical results for tsunami wave height and velocity are very close match to the real physical phenomenon of tsunami.

• Application of satellite derived bathymetry surveying as an alternate to overcome complexity of hydrographic surveying and other bathymetric techniques. • Account of satellite derived bathymetry for five decades and its systematic classification scheme grounded in research philosophy. • SDB approaches, models, methods and techniques along with chronological development of SDB algorithms, application studies, their accuracy and errors in retrieval. • A matrix of prerequisite satellite data, in-situ data resolution, methods and algorithms of SDB based on level of accuracy needs to be achieved for future researchers. • Operationalisation of satellite SDB products in coastal water. The number of civilian, commercial and military applications are dependant on accurate knowledge of bathymetry of coastal regions. Conventionally, hydrographic surveying methods are used for bathymetric surveys carried by ship-based acoustic systems, but needs high-cost resources. Space technology has provided a cost-effective alternate means for charting near shore and inaccessible waters. The optical satellite data have capabilities to offer alternate solution in near-shore region, which has been researched for past 50 years, using evolving algorithms to estimate Satellite Derived Bathymetry (SDB). However, there is no agreement on use of terms like approach, model, method and techniques, which have been used varyingly and interchangeably as per context of SDB research. This paper suggests a classification scheme for SDB algorithms which is also applicable to other Marine Remote Sensing studies. In this paper, based on literature available on SDB for the past five decades, an insight on SDB classification has been offered grounded in research philosophy. The SDB approaches, models, methods and techniques have been elaborated with chronological development, along with SDB studies based on them, their accuracy and errors in SDB retrieval. We have suggested a matrix of prerequisite satellite data, in-situ data resolution, methods and algorithms of SDB based on level of accuracy needs to be achieved, which will guide future researchers to select one as per their context of research.

• Here, the (2+1)- dimensional extended Calogero-Bogoyavlenskii-Schiff equation (ECBS) is considered to obtain the travelling wave solutions of the nonlinear problems. • Moreover, relatively new method is applied to find the new exact analytical solutions of the ECBS equation, where some of the results are discussed. • Also, the obtained solutions show the different physical natures like periodic and soliton waves in various conditions, which have been presented in this paper. In this paper, the new travelling wave solutions of the (2+1)-dimensional extended Calogero-Bogoyavlenskii-Schiff (ECBS) equation are investigated. The main aim of this work is to find the new exact solutions with the aid of relatively new ( G ′ G ′ + G + A ) -expansion method. Moreover, the physical interpretation of the nonlinear phenomena is reported through the exact solutions, which indicate the efficacy of the proposed method. Furthermore, the recovered solutions are periodic and solitary wave solutions which are presented graphically.

• Solution for the fractional potential KdV and Benjamin equations using has been investigated by the help of q -HATM. • To validate the demonstrated solution procedure is accurate and reliable; the numerical simulations have been conducted for both equations. • The physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order. • The projected technique offers two parameters n and ℏ , with the assist of these parameter nature of obtained solution is presented. In this paper, we find the solutions for fractional potential Korteweg–de Vries (p-KdV) and Benjamin equations using q -homotopy analysis transform method ( q -HATM ) . The considered method is the mixture of q -homotopy analysis method and Laplace transform, and the Caputo fractional operator is considered in the present investigation. The projected solution procedure manipulates and controls the obtained results in a large admissible domain. Further, it offers a simple algorithm to adjust the convergence province of the obtained solution. To validate the q -HATM is accurate and reliable, the numerical simulations have been conducted for both equations and the outcomes are revealed through the plots and tables. Comparison between the obtained solutions with the exact solutions exhibits that, the considered method is efficient and effective in solving nonlinear problems associated with science and technology.

• The paper aimed to study the flow structure originated from two cylinders at different angular position angles 0˚, 30˚, 45˚, 60˚, 90˚ of upstream cylinder w.r.t the downstream cylinder. • Finite Volume Method was solved by Navier -Stokes equation and energy equation to extract flow analysis at Reynolds Number of 200. • The flow parameters such as drag coefficient, Lift coefficient, Nusselt number and Strouhal number variations are analysed. • It is found that the maximum heat transfer is observed for 45˚ downwind cylinder. A numerical investigation is carried out for the different orientations of a circular cylinder where the upstream cylinder moves with varying angles α = 0˚, 30˚, 45˚, 60˚, 90˚ with respect to the downstream cylinder in a fixed position at a gap ratio ( L/D ) of 3 where L is the distance from centre to centre between cylinders and D is the diameter of the cylinder. Reynolds number ( Re ), is based on cylinder diameter, is kept constant at 200 for all the cases with air as the working medium. The vortex shedding formation is analysed when the vortices from the upwind cylinder interact with the downwind cylinder and give impressive flow patterns. It is observed that the drag and lift coefficients increase for both the cylinders with change in angular position. Strouhal number is calculated with the help of Fast Fourier transformation (FFT) of vorticity magnitude and evaluated for each case. It is observed that the effect of upwind cylinder on downwind cylinder is significant for α = 30˚, 45˚, 60˚ orientation cases and vortex shedding frequency increases for these cases. Further, the investigation is extending on the laminar forced convection heat transfer performance and variation of Nusselt number for the cylinders. Maximum heat transfer is observed for the downwind cylinder at α = 45˚. In contrast, the local Nusselt number does not vary much for the cylinders except for the downwind cylinder at α = 0˚.

• Consider a fiber-reinforced orthotropic thermoelastic rotating medium under three-phase-lag model. • Reflection phenomena is studied. • Effects of different parameters on reflection coefficients have been depicted graphically. • The expressions of energy ratios have also been obtained in explicit form and are shown graphically. • The sum of energy ratios is also verified and it is equal to unity at each angle of incidence. The present article is concerned with the propagation of plane waves in a homogeneous, fiber-reinforced orthotropic thermoelastic rotating half-space in the context of three-phase-lag model. There exist three coupled waves, namely, quasi-longitudinal P -wave ( q P ) , quasi-longitudinal thermal wave ( q T ) and quasi-transverse wave ( q S V ) in the medium. The reflection coefficients are computed numerically with the help of MATLAB programming and are depicted graphically to show the effects of rotation, fiber-reinforcement and phase lag parameters. The expressions of energy ratios have also been obtained in explicit form and are shown graphically as functions of angle of incidence. It has been verified that during reflection phenomena, the sum of energy ratios is equal to unity at each angle of incidence. Effect of anisotropy is depicted on velocities of various reflected waves. Some particular cases of interest have also been inferred from the present investigation.