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The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics.

The authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods.

First, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions.

The application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.

The purpose of this paper is to consider a steady two-dimensional magneto-hydrodynamic (MHD) Falkner-Skan boundary layer flow of an incompressible viscous electrically fluid over a permeable wall in the presence of a magnetic field.

The governing equations of MHD Falkner-Skan flow are transformed into an initial values problem of an ordinary differential equation using the Lie symmetry method which are then solved by He's variational iteration method with He's polynomials.

The approximate solution is compared with the known solution using the diagonal Pad’e approximants and the geometrical behavior for the values of various parameters. The results reveal the reliability and validity of the present work, and this combinational method can be applied to other nonlinear boundary layer flow problems.

In this paper, an approximate analytical solution of the MHD Falkner-Skan flow problem is obtained by combining the Lie symmetry method with the variational iteration method and He's polynomials.

The purpose of this paper is to concern with introducing symmetry analysis to the extended Sakovich equation.

The newly developed Sakovich equation has been handled by using the Lie symmetries via using the Lie group method.

The developed extended Sakovich model exhibit symmetries and invariant solutions.

The present study is to address the two main motivations: the study of symmetry analysis and the study of soliton solutions of the extended Sakovich equation.

The work introduces symmetry analysis to the Painlevé-integrable extended Sakovich equation.

The work presents useful symmetry algorithms for handling new integrable equations.

The paper presents an original work with symmetry analysis and shows useful findings.

The purpose of this paper is to discuss the homoclinic breathe-wave solutions and the singular periodic solutions for (2 + 1)-dimensional generalized shallow water wave (GSWW) equation.

The Hirota bilinear method, the Lie symmetry method and the non-Lie symmetry method are applied to the (2 + 1)D GSWW equation.

A reduced (1 + 1)D potential KdV equation can be derived, and its soliton solutions are also presented.

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

These results are very useful for investigating some localized geometry structures of dynamical behaviors and enriching dynamical features of solutions for the higher dimensional systems.

This paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a permeable wall in the presence of a magnetic field.

Using the Lie group approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration. Using these suitable similarity transformations, the governing partial differential equations are reduced to linear and nonlinear ordinary differential equations (ODEs). Further, Haar wavelet approach is applied to the reduced ODE under the subalgebra 4.1 for constructing numerical solutions of the flow problem.

A new type of solutions was obtained of the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis.

To find a solution for the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis is a new approach for fluid problems.

The purpose of this paper is to find new non-traveling wave solutions and study its localized structure of Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation.

The authors apply the Lie group method twice and combine with the Exp-function method and Riccati equation mapping method to the (2+1)-dimensional CDGKS equation.

The authors have obtained some new non-traveling wave solutions with two arbitrary functions of time variable.

As non-linear evolution equations is characterized by rich dynamical behavior, the authors just found some of them and others still to be found.

These results may help the authors to investigate some new localized structure and the interaction of waves in high-dimensional models. The new non-traveling wave solutions with two arbitrary functions of time variable are obtained for CDGKS equation using Lie group approach twice and combining with the Exp-function method and Riccati equation mapping method by the aid of Maple.

The purpose of this paper is to highlight the effect of combined heat and mass transfer characteristics of magnetohydrodynamic (MHD) free convection flow of an electrically conducting Newtonian fluid on circular cylinder with uniform heat/mass flux, taking into consideration the effects of uniform transverse magnetic field and thermal radiation.

An analysis is performed to study the momentum, combined heat and mass transfer characteristics of MHD free convection flow past a circular cylinder surface under the effect of thermal radiation with uniform heat and mass flux. By using Lie group method, the infinitesimal generators of governing equations are calculated. Using the resulting generators for the boundary value problem, the equations are transformed into an ordinary differential system. Numerical solutions of the outcoming non‐linear differential equations are found by using a combination of a Runge–Kutta algorithm and shooting technique.

Application of a magnetic field normal to the flow of an electrically conducting fluid gives rise to a resistive force that acts in the direction opposite to that of the flow. This resistive force tends to slow down the motion of the fluid along the cylinder and causes increases in its temperature and concentration and hence the respective changes in the wall shear stress, local Nusselt and Sherwood numbers as the magnetic parameter, respectively are changed with various values of angle which is measured in degrees from the front stagnation point on the surface. It is noted that these coefficients reduced as the magnetic parameter increases. Also, the effect of thermal radiation works as a heat source and so the quantity of heat added to the fluid increases, therefore the local Nusselt number reduced as the radiation parameter increases.

An analysis is performed to study the momentum, combined heat and mass transfer characteristics of MHD free convection flow of an electrically conducting Newtonian fluid on circular cylinder with uniform heat/mass flux with the effects of uniform transverse magnetic field and thermal radiation.

This paper provides a very useful source of coefficient of heat and mass transfer values for engineers planning to transfer heat and mass by using electrically conducting gases with uniform heat/mass flux.

The combined heat and mass transfer of an electrically conducting gases on free convection flow in the presence of magneto and thermal radiation effects are investigated and can be used by different engineers working on industry, geothermal, geophysical, technological and engineering applications.

The purpose of this paper is to introduce a variety of new completely integrable Calogero–Bogoyavlenskii–Schiff (CBS) equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for each of the developed models.

The newly developed models with time-dependent coefficients have been handled by using the simplified Hirota’s method. Moreover, multiple complex soliton solutions are derived by using complex Hirota’s criteria.

The developed models exhibit complete integrability, for specific determined functions, by investigating the compatibility conditions for each model.

The paper presents an efficient algorithm for handling integrable equations with analytic time-dependent coefficients.

The work presents new integrable equations with a variety of time-dependent coefficients. The author showed that integrable equations with time-dependent coefficients give real and complex soliton solutions.

This study presents useful algorithms for finding and studying integrable equations with time-dependent coefficients.

The paper gives new integrable CBS equations which appear in propagation of waves and provide a variety of multiple real and complex soliton solutions.

The purpose of this paper is to construct the algebraic traveling wave solutions of the (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsve (KdV-Z-K) equation, which can be usually used to express shallow water wave phenomena.

The authors apply the planar dynamical systems and invariant algebraic cure approach to find the algebraic traveling wave solutions and rational solutions of the (3 + 1)-dimensional modified KdV-Z-K equation. Also, the planar dynamical systems and invariant algebraic cure approach is applied to considered equation for finding algebraic traveling wave solutions.

As a result, the authors can find that the integral constant is zero and non-zero, the algebraic traveling wave solutions have different evolutionary processes. These results help to better reveal the evolutionary mechanism of shallow water wave phenomena and find internal connections.

The paper presents that the implemented methods as a powerful mathematical tool deal with (3 + 1)-dimensional modified KdV-Z-K equation by using the planar dynamical systems and invariant algebraic cure.

By considering important characteristics of algebraic traveling wave solutions, one can understand the evolutionary mechanism of shallow water wave phenomena and find internal connections.

To the best of the authors’ knowledge, the algebraic traveling wave solutions have not been reported in other places. Finally, the algebraic traveling wave solutions nonlinear dynamics behavior was shown.

The purpose of this paper is to present a modified form of trigonometric cubic B-spline (TCB) collocation method to solve nonlinear Fisher’s type equations. Taylor series expansion is used to linearize the nonlinear part of the problem. Five examples are taken for analysis. The obtained results are better than those obtained by some numerical methods as well as exact solutions. It is noted that the modified form of TCB collocation method is an economical and efficient technique to approximate the solution PDEs. The authors also carried out the stability analysis which proves that the method is unconditionally stable.

The authors present a modified form of TCB collocation method to solve nonlinear Fisher’s type equations. Taylor series expansion is used to linearize the nonlinear part of the problem. The authors also carried out the stability analysis.

The authors found that the proposed method results are better than those obtained by some numerical methods as well as exact solutions. It is noted that the modified form of TCB collocation method is an economical and efficient technique to approximate the solution PDEs.

The authors propose a new method, namely, modified form of TCB collocation method. In the authors’ best knowledge, aforesaid method is not proposed by any other author. The authors used this method to solve nonlinear Fisher’s type equations and obtained more accurate results than the results obtained by other methods.