Power Series Solution Research Articles

Lie symmetries, exact solutions and conservation laws of time fractional coupled (2+1)-dimensional nonlinear Schrödinger equations

ABSTRACT In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem.

Lie symmetries, exact solutions and conservation laws of time fractional Boussinesq–Burgers system in ocean waves

In this paper, the Lie symmetry analysis method is applied to the time-fractional Boussinesq–Burgers system which is used to describe shallow water waves near an ocean coast or in a lake. We obtain all the Lie symmetries admitted by the system and use them to reduce the fractional partial differential equations with a Riemann–Liouville fractional derivative to some fractional ordinary differential equations with an Erdélyi–Kober fractional derivative, thereby getting some exact solutions of the reduced equations. For power series solutions, we prove their convergence and show the dynamic analysis of their truncated graphs. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Auto-Bäcklund transformation and exact solutions for a new integrable (2+1)-dimensional shallow water wave equation

Shallow water waves (SWWs) are often used to describe water flow and wave movement in shallow water areas. The article introduces a novel (2 + 1)-dimensional SWW equation. We prove that the equation is integrable and obtain an auto-Bäcklund transformation by truncating Painlevé expansion. Using the bilinear form of the equation, a new auto-Bäcklund transformation and some exact solutions are obtained. Besides, a convergent power series solution is derived using Lie symmetry method. These exact solutions can enrich mathematical modeling and help us understand nonlinear wave phenomena. Finally, conserved vectors are derived.

Vibration analysis of piping system coupled with conical-cylindrical combined shells

Fluid-filled piping system are widely used in power cooling, fuel transportation and underwater vessel, which not only produce vibration along the system, but also transmit to the connecting structure to generate noise in ambient environment. Most investigations have concentrated on either the vibration characteristics of piping systems or of combined shells separately. In this work, an analytical impedance synthesis method (ISM) is proposed to analyze the vibration characteristics of a pipe-hull coupled system, in which the hull can be simplified to a combined conical-stiffened cylindrical-hemisphere shell. The piping system is modeled by fluid-filled beams in three-dimensional configuration. Wave function and power series solution are used to describe the displacements of cylindrical and conical shell respectively. Based on exact analytical solutions, the impedance matrix of each segment is established separately and assembled in the global coordinate by displacement continuity and force balance boundary conditions. Forced responses are compared with the experimental results to verify the correctness of the method.

Lie symmetries, exact solutions and conservation laws of (2+1)-dimensional time fractional cubic Schrödinger equation

Abstract In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators.

Analysis of scalar fields with series convolution

Wave equations for some curved spacetimes may involve functions that prevent a solution in a closed form. In some cases, these functions can be eliminated by transformations and the solutions can be found analytically. In the cases where such transformations are not available, the infinite series expansions of these functions can be convoluted with the power series solution ansatz. We study such an example where the solution is based on a special function.

A NEW ALGORITHM FOR GENERATING POWER SERIES SOLUTIONS FOR A BROAD CLASS OF FRACTIONAL PDES: APPLICATIONS TO INTERESTING PROBLEMS

This research offers a precise analytical solution for initial value problems of both linear and nonlinear fractional order partial differential equations. A new approach called the limit residual function method is developed and utilized to generate a series solution for the equations. The key tools of this method are the concepts of power series, residual function, and the limit at zero. The new method is characterized by the speed of determining the series solution coefficients and the limited mathematical operations used. The study examines five significant applications of fractional partial differential equations using this new method. To validate the accuracy of the results, they are compared with exact solutions in the classical case for the discussed applications. We conclude that the proposed approach is straightforward, effective, and successful in solving linear and nonlinear fractional differential equations.

Symmetry analysis, optimal system, conservation laws and exact solutions of time-fractional diffusion-type equation

In Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys. 17(01) (2020) 2050013], the generalized time-fractional diffusion equation [Formula: see text] is studied by the symmetry analysis method. Liu et al. obtained two group generators [Formula: see text] [Formula: see text] and only one trivial solution [Formula: see text] for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case [Formula: see text], we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically.

Power series solution for fractal differential equations

Power series solution for fractal differential equations

Invariant analysis, invariant subspace method and conservation laws of the (2+1)-dimensional mixed fractional Broer–Kaup–Kupershmidt system

In this paper, we investigate the (2+1)-dimensional mixed fractional Broer–Kaup–Kupershmidt system (MFBKKS) with Riemann–Liouville time fractional derivative and integer order y-derivative. This system models dispersive and nonlinear long gravity waves propagating in shallow water in two horizontal directions with time memories. Lie symmetry analysis and invariant subspace method are distinctly employed to construct the exact solutions to the MFBKKS. Initially, the Lie algebras admitted by MFBKKS are obtained with the help of Lie symmetry analysis. Then, we establish the commutative table, adjoint relations and adjoint transformation matrix. Specifically, the one-dimensional optimal system is established correspondingly, and symmetry reduction is performed. In particular, (2+1)-dimensional MFBKKS is reduced to (1+1)-dimensional mixed fractional system using Erdélyi–Kober fractional differential operator. Further, the power series solution of MFBKKS is constructed via power series method. By applying invariant subspace method, we obtain more exact solutions of MFBKKS. In addition, the conservation laws are derived by new Noether theorem. Finally, the three-dimensional diagrams of some obtained solutions are demonstrated utilizing Matlab for visualization, and some of the calculations were verified using computational packages Maple for symbolic computation.

Exact solution for mode shapes of cracked nonuniform axially functionally graded beams

This paper established the exact formulas for mode shapes of cracked nonuniform axially functionally graded (AFG) beams. The formula of the mode shape of cracked nonuniform AFG beam is derived in the form of a power presented as a recurrent relation. The power series solution of cracked beam is the same with the intact beam where the effect of cracks contributes to only the first four coefficients of the power series. The derivation of the formula of mode shapes is presented in details. Numerical simulations show that these effects on the natural frequency and mode shape are quite significant. The natural frequency and mode shape are more sensitive to cracks located at positions with low elastic modulus, small cross section and high mass density. For the case of tapered AFG cantilever beam, it is important to consider not only cracks located close to the fixed end but also cracks located close to the free end, especially when the free end has lower elastic modulus and higher mass density.

Exact solution for heat transfer across the Sakiadis boundary layer

We consider the problem of convective heat transfer across the laminar boundary layer induced by an isothermal moving surface in a Newtonian fluid. In a previous work [Barlow et al., “Exact and explicit analytical solution for the Sakiadis boundary layer,” Phys. Fluids 36, 031703 (2024)], an exact power series solution was provided for the hydrodynamic flow, often referred to as the Sakiadis boundary layer. Here, we utilize this expression to develop an exact solution for the associated thermal boundary layer as characterized by the Prandtl number (Pr) and local Reynolds number along the surface. To extract the location-dependent heat transfer coefficient (expressed in dimensionless form as the Nusselt number), the dimensionless temperature gradient at the wall is required; this gradient is solely a function of Pr and is expressed as an integral of the exact boundary layer flow solution. We find that the exact solution for the temperature gradient is computationally unstable at large Pr, and a large Pr expansion for the temperature gradient is obtained using Laplace's method. A composite solution is obtained, which is accurate to O(10−10). Although divergent, the classical power series solution for the Sakiadis boundary layer—expanded about the wall—may be used to obtain all higher-order corrections in the asymptotic expansion. We show that this result is connected to the physics of large Prandtl number flows where the thickness of the hydrodynamic boundary layer is much larger than that of the thermal boundary layer. The present model is valid for all Prandtl numbers and attractive for ease of use.

Optimized technique and dynamical behaviors of fractional Lax and Caudrey–Dodd–Gibbon models modelized by the Caputo fractional derivative

The presented paper aims to investigate, examine, and analyze the nonlinear time-fractional evolution partial differential equations (TFNE-PDEs) in the sense of Caputo essential in numerous nonlinear wave propagation phenomena. To achieve this, the Laplace-residual power series method (L-RPSM) is used to construct approximate Laplace-residual power series solutions (AL-RPSSs) for fifth-order Korteweg–de Vries PDEs (KdV-PDEs). The L-RPSM provides analytical solutions of the dynamic wavefunction of several equations including time-fractional Lax PDE (TFL-PDE), and time-fractional Caudrey–Dodd–Gibbon PDEs (TFCDG-PDEs). The theoretical and numerical consequences of the models are discussed, and error’s analysis of the method are discussed. The outcomes are introduced in 2D and 3D figures, and dynamic behaviors of parameters are discussed for distinct values of α. Moreover, the accuracy of this method is demonstrated by comparing it with results obtained using other methods. The results show that the suggested iterative approach is a suitable tool with computational efficiency for long-wavelength solutions of nonlinear time-fractional PDEs in various phenomena.

Lie Symmetry Analysis, Power Series Solutions and Conservation Laws of (2+1)-Dimensional Time Fractional Modified Bogoyavlenskii–Schiff Equations

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations, which is an important model in physics. The one-dimensional optimal system composed by the obtained Lie symmetries is utilized to reduce the system of (2+1)-dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to the system of (1+1)-dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative. Then the power series method is applied to derive explicit power series solutions for the reduced system. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Similarity reduction, group analysis, conservation laws, and explicit solutions for the time-fractional deformed KdV equation of fifth order

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of [Formula: see text] which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail.

Investigating radiative heat transfer, varied wall thickness, and slip effects on Casson nanofluid flow over a stretched sheet with heat source

ABSTRACT Understanding the intricate interplay between variable fluid properties such as slip and thermal conductivity when flowing over porous surfaces is of utmost importance for a wide range of engineering applications. This investigation delves into this uncharted territory, connecting the analytical capabilities of HAM (Homotopy Analysis Method) to reveal captivating insights into the impact of these variables on the dynamics of Casson nanofluid flow. Besides, we documented the flow aspects which include thermal radiation, heat source, variable wall thickness and chemical reaction. We alter the partial differential flow-related conditions into nonlinear ordinary ones employing the similarity transformation approach. Then, using a popular semi-analytical technique known as the Homotopy Analysis Method (HAM), we were able to untangle them. This method yields to power series solutions to nonlinear differential equations. To illustrate the impact of the velocity, temperature and concentration profiles, parametric research has been done using tables and diagrams. In the limiting sense, the numerical results of our methodology are in great association with the outcomes of previous research. Finally, it is noted that higher values of the velocity slip parameter cause an enhancement in fluid velocity, while escalating values of the thermal slip parameter cause a decline in temperature distribution.

On a family of nonzero solutions to the heat equation ut = △u on ℝn+1 which vanish on an arbitrary n-dimensional hyperplane P ⊂ ℝn+1

Motivated by the paper Rosenbloom–Widder [A temperature function which vanishes initially, Amer. Math. Mon. 65(8) (1958) 607–609], we construct a more general family of nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data. These solutions are analytically more manageable than the well-known Tychonoff power series solution. Nonzero solutions to the [Formula: see text]-dimensional heat equation [Formula: see text] on [Formula: see text] with zero initial data (i.e. vanishing on the hyperplane [Formula: see text]) can be easily obtained as a consequence of the examples on [Formula: see text] Finally, given an arbitrary hyperplane [Formula: see text], we can construct a family of nonzero solutions which vanish on [Formula: see text]

On the generalized time fractional reaction–diffusion equation: Lie symmetries, exact solutions and conservation laws

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically.

Invariant analysis, approximate solutions, and conservation laws for the time fractional higher order Boussinesq–Burgers system

In this paper, we employ the time fractional higher order nonlinear Boussinesq–Burgers system to investigate shallow water waves based on the Riemann–Liouville and Caputo fractional partial derivatives. The Lie algebra of infinitesimal generators associated with the symmetry groups that leave the studied system invariant is systematically constructed, and the similarity reductions are established. Furthermore, conserved quantities of the system under consideration are formulated. Next, the power series solution, including the convergence analysis, is obtained. Based on the fractional Sumudu–Adomian decomposition method, some approximate solutions are derived. Finally, in order to show the efficiency of the considered methods, the effect of the fractional order, as well as the dynamical behavior of solutions, numerical validation and graphical representations are designed.

Asymptotic existence theorem for formal power series solutions of singularly perturbed linear q-difference equations

Asymptotic existence theorem for formal power series solutions of singularly perturbed linear q-difference equations