Power Series Solution Research Articles

Lie symmetry analysis, conservation laws and analytical solutions for chiral nonlinear Schrödinger equation in (2 + 1)-dimensions

In this work, we consider the chiral nonlinear Schrödinger equation in (2 + 1)-dimensions, which describes the envelope of amplitude in many physical media. We employ the Lie symmetry analysis method to study the vector field and the optimal system of the equation. The similarity reductions are analyzed by considering the optimal system. Furthermore, we find the power series solution of the equation with convergence analysis. Based on a new conservation law, we construct the conservation laws of the equation by using the resulting symmetries.

Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients

The Lie point symmetries are reported by performing the Lie symmetry analysis to the Ablowitz-Kaup-Newell-Suger (AKNS) equation with time-dependent coefficients. In addition, the optimal system of one-dimensional subalgebras is constructed. Based on this optimal system, several categories of similarity reduction and some new invariant solutions for the equation are obtained, which include power series solutions and travelling and non-traveling wave solutions.

New approaches for the solution of space-time fractional Schr\xf6dinger equation

The aim of this study is to establish the solution of the time-space fractional Schrödinger equation subject to initial and boundary conditions which has many applications in science such as nonlinear optics, plasma physics, super conductivity, based on the residual power series method (RPSM). We first apply suitable transformations to make the order of one of the fractional derivatives integer to implement the RPSM easily to construct the fractional power series solution. The method proposed in this article gives highly encouraging results. Illustrative examples show that this method is compatible with solving such fractional differential equations.

Matrix-valued Bratu equation and the exact solution of its initial value problem

A matrix-valued extension of the Bratu equation is defined. For its initial value problem, the exponential matrix solution and power series solution are provided.

Analysis of the evolution equation of a hyperbolic curve flow via Lie symmetry method

In this paper, based on the classical symmetry method, the group-invariant solutions of the evolution equation of a hyperbolic curve flow are investigated. The optimal system of the obtained symmetries is found, and the reduced equations and exact solutions of the evolution equation are discussed. Then explicit solutions are obtained by the power series method. In addition, the convergence of the power series solutions is proved. The objective shapes of the solutions of the evolution equation are performed.

Lie symmetry analysis, conservation laws and some exact solutions of the time-fractional Buckmaster equation

This paper systematically investigates the Lie symmetry analysis of the time-fractional Buckmaster equation in the sense of Riemann–Liouville fractional derivative. With the aid of infinitesimal symmetries, this equation is transformed into a nonlinear ordinary differential equation of fractional order (FODE), where the fractional derivatives are in Erdelyi–Kober sense. The reduced FODE is solved with the explicit power series method and some figures for the obtained power series solutions are also depicted. Finally, Ibragimov’s method and Noether’s theorem have been employed to conclude the conservation laws of this equation.

A Non-Singular Analytical Technique for Reinforced Non-Circular Holes in Orthotropic Laminate

The intricacy in Lekhnitskii’s available single power series solution for stress distribution around hole edge for both circular and noncircular holes represented by a hole shape parameter ε is decoupled by introducing a new technique. Unknown coefficients in the power series in ε are solved by an iterative technique. Full field stress distribution is obtained by following an available method on Fourier solution. The present analytical solution for reinforced square hole in an orthotropic infinite plate is derived by completely eliminating stress singularity that depends on the concept of stress ratio. The region of validity of the present analytical solution on reinforcement area is arrived at based on a comparison with the finite element analysis. The present study will also be useful for deriving analytical solution for orthotropic shell with reinforced noncircular holes.

Lie symmetry analysis and traveling wave solutions of equal width wave equation

We obtained the power series solution and the traveling wave solutions of equal width wave equation by using the Lie symmetry method. The fundamental idea behind the symmetry transformation method is that it reduces one independent variables in a system of PDEs by utilizing Lie symmetries and surface invariance condition. We first obtained the infinitesimals and commutation table with the help of MAPLE software. Lie symmetry transformation method (STM) has been applied on EWW equation and converted it into various nonlinear ODEs. Then, the tanh method and the power series method have been applied for solving the reduced nonlinear ordinary differential equations (ODEs). Convergence of the power series solutions has also been shown.

Asymptotic Approximant for the Falkner–Skan Boundary Layer Equation

Summary We demonstrate that the asymptotic approximant applied to the Blasius boundary layer flow over a flat plat (Barlow et al., Q. J. Mech. Appl. Math. 70 (2017) 21–48.) yields accurate analytic closed-form solutions to the Falkner–Skan boundary layer equation for flow over a wedge having angle $\beta\pi/2$ to the horizontal. A wide range of wedge angles satisfying $\beta\in[-0.198837735, 1]$ are considered, and the previously established non-unique solutions for $\beta<0$ having positive and negative shear rates along the wedge are accurately represented. The approximant is used to determine the singularities in the complex plane that prescribe the radius of convergence of the power series solution to the Falkner–Skan equation. An attractive feature of the approximant is that it may be constructed quickly by recursion compared with traditional Padé approximants that require a matrix inversion. The accuracy of the approximant is verified by numerical solutions, and benchmark numerical values are obtained that characterize the asymptotic behavior of the Falkner–Skan solution at large distances from the wedge.

Optimal maneuver penetration strategy based on power series solution of miss distance

Optimal maneuver penetration strategy based on power series solution of miss distance

An efficient computational scheme for nonlinear time fractional systems of partial differential equations arising in physical sciences

In this paper, we broaden the utilization of a beautiful computational scheme, residual power series method (RPSM), to attain the fractional power series solutions of nonhomogeneous and homogeneous nonlinear time-fractional systems of partial differential equations. This paper considers the fractional derivatives of Caputo-type. The approximate solutions of given systems of equations are calculated through the utilization of the provided initial conditions. This iterative scheme generates the fast convergent series solutions with conveniently determinable components. The implementation of this numerical scheme clearly exhibits its effectiveness, reliability and easiness regarding the procedure of the solution, as well as its better approximation. The repercussions of the fractional order of Caputo derivatives on solutions are depicted through graphical presentations for various particular cases.

Lie Symmetry Analysis and Invariant Solutions of a Nonlinear Fokker-Planck Equation Describing Cell Population Growth

In this paper, a complete Lie symmetry analysis is performed for a nonlinear Fokker-Planck equation for growing cell populations. Moreover, an optimal system of one-dimensional subalgebras is constructed and used to find similarity reductions and invariant solutions. A new power series solution is constructed via the reduced equation, and its convergence is proved.

Vectorial Iterative Fractional Laplace Transform Method for the Analytic Solutions of Fractional Cauchy-Riemann Systems Partial Differential Equations

The present study aims to obtain infinite fractional power series solution vectors of fractional Cauchy-Riemann systems equations with initial conditions by the use of vectorial iterative fractional Laplace transform method (VIFLTM). The basic idea of the VIFLTM was developed successfully and applied to four test examples to see its effectiveness and applicability. The infinite fractional power series form solutions were successfully obtained analytically. Thus, the results show that the VIFLTM works successfully in solving fractional Cauchy-Riemann system equations with initial conditions, and hence it can be extended to other fractional differential equations.

The solutions of multiplicative Hermite differential equation and multiplicative Hermite polynomials

In this work, we give the power series solutions around an ordinary point, in the case of variable coefficients, homogeneous multiplicative differential equations of 2nd order. Further, we introduce the multiplicative Hermite differential equations, multiplicative Hermite polynomials and basic properties of these polynomials.

Pressure build-up analysis in the flow regimes of the CO2 sequestration problem

In this work we analyse theoretically and numerically the pressure build-up on the cap rock of a saline aquifer during CO2 injection in all flow regimes. Flow regimes are specific regions of the parameter space representing the mathematical spread of the plume. The parameter space is defined in terms of the CO2-to-brine relative mobility λ and the buoyancy parameter Γ. In addition to the known asymptotic self-similar solutions for low buoyancy regimes, we introduce two novel ones for the high buoyancy regimes via power series solutions. Explicit results for the peak pressure value on the cap, which arises in the vicinity of the well, are derived and discussed for all flow regimes. The analytical results derived are then applied for cap integrity considerations in six test cases of CO2 geological storage from the PCOR partnership, most of which correspond to high buoyancy conditions. The validity of the self-similar solutions which are late time asymptotics, is verified with CFD numerical simulations with a commercial software. The comparison between the self-similar solutions and CFD for the pressure estimations are in excellent agreement and the self-similar solutions are valid for typical injection durations even for early times.

Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

Given a first order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial values, classified in terms of the number of distinct formal power series solutions extending them. In addition, if a particular initial value is given, we present a second algorithm that computes all the formal power series solutions, up to a suitable degree, corresponding to it. Furthermore, when the ground field is the field of the complex numbers, we prove that the computed formal power series solutions are all convergent in suitable neighborhoods.

The generalized convective Cahn–Hilliard equation: Symmetry classification, power series solutions and dynamical behavior

We first perform a complete Lie symmetry classification of the generalized convective Cahn–Hilliard equation. Then using the obtained symmetries, we mainly study the convective Cahn–Hilliard equation, of which a new power series solution is constructed. In particular for the crystal surface growth processes, the truncated series solution shows that the surface structures include peaks and valleys, and can exhibit different evolution trends with the driving force varying from compressive force to tensile force. Moreover, there exist several critical points for the driving force, where the surface configurations take the jump changes and show different features on the both sides of such critical points. According to the effects of driving forces, we analyze the dynamical features of crystal growth.

Series solutions for the Laguerre and Lane-Emden fractional differential equations in the sense of conformable fractional derivative

In the present paper, we use efficient and simple algorithms of the fractional power series and Adomain polynomial methods that provide effective tools for solving such linear and nonlinear fractional differential equations in the sense of conformable derivative. The first method succeeded in finding the fractional power series solution for the Laguerre linear fractional differential equation with some important special cases. The second method is applied to predict and construct the fractional Adomain approximation solution of the general conformable Lane-Emden fractional differential equation with some interesting linear and nonlinear special cases with their graphs. Finally, the results show that the Adomain approximation solutions of those problems are stable at different values of α and assure that this way can be applied successfully for solving many linear and nonlinear fractional differential equations in conformable sense.

Explicit Solutions of Coupled Time Fractional Kaup-Boussinesq Equation with Weak Dispersion

In the current article, we consider a system of time fractional Kaup-Boussinesq shallow water equations. Using Lie group analysis, we obtain the symmetry group of transformations which reduces the system of fractional partial differential equations (FPDEs) to system of fractional ordinary differential equations (FODEs). Further, we investigate the exact explicit group invariant solution as well as power series solution of the given system of equations. Next the physical significance of the group invariant solution under the influence of fractional order is studied graphically. Lastly, conserved vectors for the FPDEs are obtained using the conservation theorem.

Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis

Abstract In this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.