Fourth-order Nonlinear Ordinary Differential Equation Research Articles

Generalized and multi-oscillation solitons in the nonlinear Schrödinger equation with quartic dispersion.

We study different types of solitons of a generalized nonlinear Schrödinger equation (GNLSE) that models optical pulses traveling down an optical waveguide with quadratic as well as quartic dispersion. A traveling-wave ansatz transforms this partial differential equation into a fourth-order nonlinear ordinary differential equation (ODE) that is Hamiltonian and has two reversible symmetries. Homoclinic orbits of the ODE that connect the origin to itself represent solitons of the GNLSE, and this allows one to study the existence and organization of solitons with advanced numerical tools for the detection and continuation of connecting orbits. In this paper, we establish the existence of new types of connecting orbits, namely, PtoP connections from one periodic orbit to another. As we show, these global objects provide a general mechanism that generates additional families of two types of solitons in the GNLSE. First, we find generalized solitons with oscillating tails whose amplitude does not decay but reaches a nonzero limit. Second, PtoP connections in the zero energy level can be combined with EtoP connections from the origin to a selected periodic orbit to create multi-oscillation solitons; their characterizing property is to feature several episodes of different oscillations in between decaying tails. As is the case for solitons that were known previously, generalized solitons and multi-oscillation solitons are shown to be an integral part of the phenomenon of truncated homoclinic snaking.

Similarity reduction and multiple novel travelling and solitary wave solutions for the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients

In this study, the two-dimensional Bogoyavlensky–Konopelchenko equation with variable coefficients (vcBK) which describes the interaction of Riemann waves has been solved two times by using the balance technique. First, the balance procedure was used to reduce the vcBK equation into a fourth-order nonlinear ordinary differential equation and by its implementation again composed with the auxiliary method, many novel periodic, kink and solitary wave solutions were obtained for the vcBK equation. Second, the balance method was used to construct the Bäcklund transformation for the vcBK equation, by which n-soliton solutions were obtained. Finally, some figures were given to show the soliton interaction for different values of the variable coefficients.

Effects of an external constant pressure gradient on a steady incompressible laminar flow through a semi-porous annular pipe

Abstract In this paper, we examine a steady laminar flow for an incompressible fluid located in a semi porous annular pipe and subjected to a favorable constant pressure gradient applied between the two borders of the pipe. The inner wall is impermeable and the fluid is sucked or injected at the outer wall at constant and uniform velocity, orthogonally to the wall. The problem under study depends on three parameters: the pipe gap ratio, the dimensionless external pressure gradient, and the Reynolds number defined from the sum of the suction or injection velocity and the maximum Hagen–Poiseuille velocity. The conservation of mass induces the zero-divergence velocity field which allows replacing the steady-flow Navier–Stokes equations with a single equation satisfied by the stream function and called the vorticity equation. Assuming the similarity-solution hypothesis, the problem under consideration is reduced to a fourth-order nonlinear ordinary differential equation with two boundary conditions at each wall. The numerical shooting technique including the Runge–Kutta algorithm and the Newton–Raphson optimization method is applied to obtain the solution for the steady flow. For various values of the dimensionless external pressure gradient, the profiles of the velocity components are found and investigations on the wall shear stress for both walls are performed. The results obtained are discussed and physical understandings for the problem studied are derived.

Numerical treatment of squeezing unsteady nanofluid flow using optimized stochastic algorithm

Abstract In this paper, very efficient, intelligent techniques have been used to solve the fourth-order nonlinear ordinary differential equations arising from squeezing unsteady nanofluid flow. The activation functions used to develop the three models are log-sigmoid, radial basis, and tan-sigmoid. The neural network of each scheme is optimized with the interior point method (IPM) to find the weights of the networks. The confrontation of the obtained results with the numerical solutions shows good accuracy of the three schemes. The obtained solutions by utilizing the neural network technique of our variables field (velocity and temperature) are continuous contrary to the discrete form obtained by the numerical scheme.

Symmetry solutions and conservation laws of a (3+1)-dimensional generalized KP-Boussinesq equation in fluid mechanics

In this paper we analyse a (3+1)-dimensional generalized Kadomtsev-Petviashvili-Boussinesq equation, which describes the evolution of shallow water waves. This equation was formulated not long ago, by including the term uttto the generalized (3+1)-dimensional Kadomtsev-Petviashvili equation. Invoking the Lie symmetry methods we perform several symmetry reductions and reduce the equation to a fourth-order nonlinear ordinary differential equation. Consequently, this ordinary differential equation is solved and its general solution is derived connected with Weierstrass zeta function. The profiles of solutions are displayed graphically. In addition to those, we construct some travelling waves by engaging Kudryashov’s method. To complete our study we finally derive some conservation laws of the equation under consideration by appealing to the Ibragimov’s conservation theorem.

Lidstone-type problems on the whole real line and homoclinic solutions applied to infinite beams

This work provides sufficient conditions for the existence of solutions to fourth-order nonlinear ordinary differential equations with Lidstone-type boundary conditions on the real line. Using Green’s functions, we formulate a modified integral equation and correspondent integral operators, in which fixed points are the solutions of the initial problem. Moreover, it is proved that every solution of the Lidstone problem on the whole real line is an homoclinic solution.

Catalogue of existence of the multiple physical solutions of hydromagnetic flow over a stretching/shrinking sheet for viscoelastic second-grade and Walter's B fluids

In this work, the author restudied a fourth-order nonlinear ordinary differential equation that commonly appears in investigating the hydromagnetic flow of two types of viscoelastic fluids over a stretching/shrinking sheet. In this type of problem, vital question arises: what are the parameters values needed to obtain regions and numbers of the physical solutions? To answer this question, the researchers worked on suggesting/estimating the initial values, or even chose them randomly. This usually leads to wasting a lot of time, effort and maybe to untrustworthy results. Towards avoiding this rugged road, the present work answered this question decisively and definitively. In particular, an explicit and direct catalogue of existence multiple real positive solutions was in detail introduced. MATHEMATICA code was programmed and, therefore, many tables and figures of the critical values and regions were produced at different values of the controlled parameters. In addition, and maybe for the first time, the mathematical expressions of these critical values were secured. Hence, remarkable differences were detected between the existing results and those obtained in the literature, therefore, spurious physical sights were gained. In conclusion, this research determined the whole values of the parameters that are required to control the number of solutions or even not to obtain solutions at all. Therefore, the current catalogue aims to help the researchers to avoid guessing the parameters’ values on solving the current type of problem.

The Solution of the Fractional Form of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries by Analytical Techniques

In this article, two powerful techniques called variational iteration andAdomian decomposition methods are proposed to solve analytically thefractional form of the fourth order nonlinear ordinary differentialequation which reduced by similarity transformation from the basicsystem of nonlinear partial differential equations of motion of anunsteady axisymmetric flow of nonconducting, Newtonian fluidsqueezed between two circular plates with slip and no-slip boundaries.The analysis of convergence of the proposed methods is discussedthrough the absolute residual errors for various Reynolds number andvarious values of the fractional order. Comparisons between the resultsobtained by the proposed methods with those obtained by the newiterative and Picard methods are made which confirm that the proposedmethods are powerful methods and therefore suitable for solving thiskind of problems.

A class of numerical methods for the solution of fourth-order nonlinear ordinary differential equations on a graded mesh with boundary conditions of first kind

We propose difference schemes of order 2 and 3 on a graded mesh for fourth-order nonlinear ordinary differential equations (ODEs). The values of u and (du/dx) are prescribed at two boundary points. Our methods involve only 3-grid points. A graded mesh is generated by introducing a mesh ratio parameter σ. Numerical values of (du/dx) are obtained as a by-product of the proposed methods. The stability and convergence analysis are discussed in brief. The modified version of the method works well for singular problems. We have tested our methods on seven benchmark problems to demonstrate the usefulness of the proposed methods.

Suction and injection effect on flow between two plates with reference to Casson fluid model

PurposeThe purpose of this paper is to study the effect of injection and suction on velocity profile, skin friction and pressure distribution of a Casson fluid flow between two parallel infinite rectangular plates approaching or receding from each other with suction or injection at the porous plates.Design/methodology/approachThe governing Navier–Stokes equations are reduced to the fourth-order non-linear ordinary differential equation through the similarity transformations. The approximated analytic solution based on the Homotopy perturbation method is given and also compared with the classical finite difference method.FindingsFrom this study, the authors observed that the skin friction is less in non-Newtonian fluids compared to Newtonian fluids. The use of non-Newtonian fluids reduces the pressure in all the cases compared to Newtonian and hence load-carrying capacity will be more. As γ value increases velocity, skin friction and pressure decreases. When γ is fixed, it is observed that skin friction and pressure is minimum for A=0.5 and maximum when A=−0.5. The result of this study also shows that the effect of suction on the velocity profiles, pressure and skin friction is opposite to the effect of injection.Originality/valueThe present work analyzes the characteristic of non-Newtonian fluid having practical and industrial applications.

Non-monotonic blow-up problems: Test problems with solutions in elementary functions, numerical integration based on non-local transformations

We consider blow-up problems having non-monotonic singular solutions that tend to infinity at a previously unknown point. For second-, third-, and fourth-order nonlinear ordinary differential equations, the corresponding multi-parameter test problems allowing exact solutions in elementary functions are proposed for the first time. A method of non-local transformations, that allows to numerically integrate non-monotonic blow-up problems, is described. A comparison of exact and numerical solutions showed the high efficiency of this method. It is important to note that the method of non-local transformations can be useful for numerical integration of other problems with large solution gradients (for example, in problems with solutions of boundary-layer type).

Analytic Comparison of MHD Squeezing Flow in Porous Medium with Slip Condition

The aim of this paper is to compare the efficiency of various techniques for squeezing flow of an incompressible viscous fluid in a porous medium under the influence of a uniform magnetic field squeezed between two large parallel plates having slip boundary. Fourth-order nonlinear ordinary differential equation is obtained by transforming the Navier-Stokes equations. Resulting boundary value problem is solved using Differential Transform Method (DTM), Daftardar Jafari Method (DJM), Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), and Optimal Homotopy Asymptotic Method (OHAM). The problem is also solved numerically using Mathematica solver NDSolve. The residuals of the problem are used to compare and analyze the efficiency and consistency of the abovementioned schemes.

Iterative Methods for Solving the Fractional Form of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries

An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is proposed with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations of motion is reduced to a single fourth-order nonlinear ordinary differential equation. By using the basic definitions of fractional calculus, we introduced the fractional order form of the fourth-order nonlinear ordinary differential equation. The resulting boundary value fractional problems are solved by the new iterative and Picard methods. Convergence of the considered methods is confirmed by obtaining absolute residual errors for approximate solutions for various Reynolds number. The comparisons of the solutions for various Reynolds number and various values of the fractional order confirm that the two methods are identical and therefore are suitable for solving this kind of problems. Finally, the effects of various Reynolds number on the solution are also studied graphically.

Exact solitary wave and quasi-periodic wave solutions of the KdV-Sawada-Kotera-Ramani equation

In this paper we derive new exact solitary wave solutions and quasi-periodic traveling wave solutions of the KdV-Sawada-Kotera-Ramani equation by using a method which we introduce here for the first time. Firstly, we reduce the associated fourth-order nonlinear ordinary differential equation (ODE) into a solvable first-order nonlinear ODE to obtain new exact traveling wave solutions, including the solitary wave and periodic solutions. Furthermore, using the new method we derive the quasi-periodic wave solutions of this equation by assuming that the solutions of the corresponding higher-order ODE are the sum of the solutions of two solvable first-order nonlinear ODEs. This new method can be used to investigate the exact traveling wave solutions and quasi-periodic wave solutions of a general class of higher-order wave equations.

Strength Analysis of Transitional Segment of Buckling Tubing Nearby the Packer in Vertical Well

The buckling deformation and stress distribution of the tubing nearby the packer would be seriously influenced by the packer constraints. This paper focused on the Strength analysis of transitional segment. The method to calculate the buckling deformation and the equivalent stress of transitional segment were derived considering the tubing boundary and continuity condition, adopting the fourth-order nonlinear ordinary differential equations and helical buckling strength method. This study makes up for the weakness of traditional stress analysis of the buckling tubing nearby the packer in vertical well, improves the pertinence and accuracy of stress analysis of buckling tubing. The results of analysis show that with the axial pressure increasing lead to the length of the transitional segment irregular fluctuations and the maximum equivalent stress increasing. The equivalent stress would gradually reduce and finally approach a constant.

Existence and uniqueness of nonlinear deflections of an infinite beam resting on a non-uniform nonlinear elastic foundation

We consider the static deflection of an infinite beam resting on a nonlinear and non-uniform elastic foundation. The governing equation is a fourth-order nonlinear ordinary differential equation. Using the Green's function for the well-analyzed linear version of the equation, we formulate a new integral equation which is equivalent to the original nonlinear equation. We find a function space on which the corresponding nonlinear integral operator is a contraction, and prove the existence and the uniqueness of the deflection in this function space by using Banach fixed point theorem. 2010 Mathematics Subject Classification: 34A12; 34A34; 45G10; 74K10.

Effects of Boundary Conditions and Friction on Static Buckling of Pipe in a Horizontal Well

Summary A comprehensive buckling model, a group of fourth-order nonlinear ordinary differential equations, was derived by applying the principle of virtual work. Lateral friction force is included in this model. The equations were normalized to make the solutions independent of the wellbore size, type of pipe, and mud. The critical sinusoidal buckling load of tubing with different boundary conditions typically seen in drilling and well-completion applications was analyzed on the basis of the analytical solution of the linearized buckling equation. The results show that the effect of the boundary conditions can be neglected when the dimensionless length of tubing is greater than 5π. The authors further investigated the effects of friction on sinusoidal buckling by applying the principle of virtual work. The critical conditions for initiating sinusoidal buckling were determined by a group of three nonlinear equations. A perturbation solution of these nonlinear equations was obtained. It was found that the critical loads for sinusoidal buckling will increase by 30 to 70% for friction coefficients between 0.1 and 0.3. The authors also conducted an experimental study. The experimental results, including both data obtained by the authors and results published by other researchers, support the proposed model.

Non–existence of truly solitary waves in water with small surface tension

This paper considers permanent capillary–gravity waves on the free surface of a two–dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth–order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non–existence of truly solitary waves is obtained by using a contradiction on this constant.

Bäcklund transformation, lax pairs, symmetries and exact solutions for variable coefficient KdV equation

The homogeneous balance method is extended to seek for Backlund transformation, Lax pairs, non-local symmetries of variable coefficient KdV equation (VCKdVE). Then based on the Backlund transformation and general solutions of a fourth-order nonlinear ordinary differential equation, five kinds of exact solutions of VCKdVE are derived. The soliton-like solution also belongs to these solutions.

Shape of a Vapor Stem During Nucleate Boiling of Saturated Liquids

The transport processes occurring in an evaporating two-dimensional vapor stem formed during saturated nucleate boiling on a heated surface are modeled and analyzed numerically. From the heater surface heat is conducted into the liquid macro/microthermal layer surrounding the vapor stems and is utilized in evaporation at the stationary liquid–vapor interface. A balance between forces due to curvature of the interface, disjoining pressure, hydrostatic head, and liquid drag determines the shape of the interface. The kinetic theory and the extended Clausius–Clapeyron equation are used to calculate the evaporative heat flux across the liquid–vapor interface. The vapor stem shape calculated by solving a fourth-order nonlinear ordinary differential equation resembles a cup with a flat bottom. For a given wall superheat, several metastable states of the vapor stem between a minimum and maximum diameter are found to be possible. The effect of wall superheat on the shape of the vapor stem is parametrically analyzed and compared with limited data reported in the literature.