Third-order Partial Differential Equation Research Articles

Conservation laws and exact solutions of a class of non linear regularized long wave equations via double reduction theory and Lie symmetries

The conservation laws for (1+1)-dimensional non-linear generalized regularized long wave (GRLW) equation are derived via partial Noether approach after increasing its order. The GRLW equation is a third-order partial differential equation. We convert GRLW equation to fourth order equation by assuming new dependent variable v to be the derivative of original dependent variable u by setting either u=vx or u=vt. The partial Noether’s approach is then used to derive the conservation laws. The derived conserved vectors are adjusted to satisfy the divergence relationship. Finally, the conservation laws are expressed in the variable u and they constitute the conservation laws for the third-order GRLW equation. The Lie point symmetries for GRLW equation are computed. The double reduction theory based on symmetry and its associated conserved vector is utilized and two independent exact solutions are obtained. Moreover, the Lie symmetry method is used to derive an invariant solution. One of the solutions obtained by the double reduction method is the same as derived by Lie symmetry method. The second solution constructed by the double reduction theory is not obtained by the Lie symmetry method. A similar analysis is performed for regularized long wave (RLW) and modified Benjamin–Bona–Mahoney (MBBM) equations.

Loewy decomposition of linear differential equations

This paper explains the developments on factorization and decomposition of linear differential equations in the last two decades. The results are applied for developing solution procedures for these differential equations. Although the subject is more than 100 years old, it has been rediscovered as recently as about 20 years ago. A fundamental ingredient has been the easy availability of symbolic computation systems to accomplish the extensive calculations usually involved in applications; to this end the interactive website http://www.alltypes.de has been provided. Although originally only developed for ordinary equations, it has been extended to large classes of partial equations as well. In the first part Loewy’s results for ordinary equations are outlined. Thereafter those results of differential algebra are summarized that are required for extending Loewy’s theory to partial equations. In the remaining part a fairly complete discussion of second- and some third-order partial differential equations in the plane is given; it is shown that Loewy’s result remains essentially true for these equations. Finally, several open problems and possible extensions are discussed.

Conservation laws for a class of quasi self-adjoint third order equations

In this work we consider a class of third-order nonlinear partial differential equation containing two un-specified coefficient functions of the dependent variable which include various integrable and nonintegrable equations. We determine the subclasses of these equations which are self-adjoint and quasi self-adjoint. By using a general theorem on conservation laws proved by Nail Ibragimov we find conservation laws for some of these partial differential equations without classical Lagrangians.

SOLITARY TRAVELING WATER WAVES OF MODERATE AMPLITUDE

We prove the existence of solitary traveling wave solutions for an equation describing the evolution of the free surface for waves of moderate amplitude in the shallow water regime. This nonlinear third-order partial differential equation arises as an approximation of the Euler equations, modeling the unidirectional propagation of surface water waves. We give a description of the solitary wave profiles by performing a phase plane analysis and study some qualitative features of the solutions.

Group Invariant Solution for a Liquid Film on the Surface of a Sphere

When a circular jet of liquid strikes a sphere normal to the tangent plane at a point, a liquid film is formed on the surface of the sphere. This is a new problem. The flow in the liquid film is studied by means of boundary layer theory for laminar flow. The governing equations are Prandtl’s momentum boundary layer equation and the continuity equation. To solve the problem completely a conserved quantity is required as well as boundary conditions. The conserved quantity for the film of liquid on the sphere is derived with the help of a conserved vector. Two conservation laws for the system have been obtained by Naeem and Naz (Int. J. Nonlin. Sci. 7, 149 (2009)), and one of these is used to derive the conserved quantity for the liquid film. A stream function is introduced which reduces the system to a single third-order partial differential equation. The group invariant solution for this partial differential equation is constructed by considering a linear combination of its Lie point symmetries. The velocity profile of the liquid film on the sphere is investigated.

Group invariant solution for a free jet on a hemi-spherical shell

When two circular jets impinge upon each other along the axis of a hemi-spherical shell then a free jet on a hemi-spherical shell is formed. The governing equations are Prandtl’s momentum boundary layer equation and the continuity equation. The conserved quantity is required for the free jet on a hemi-spherical shell. The conserved quantity for the free jet on a hemi-spherical shell is established with the help of a conserved vector. The group invariant solution for the third-order partial differential equation for the stream function is constructed.

A second Wronskian formulation of the Boussinesq equation

A Wronskian formulation leading to rational solutions is presented for the Boussinesq equation. It involves third-order linear partial differential equations, whose representative systems are systematically solved. The resulting solutions formulas provide a direct but powerful approach for constructing rational solutions, positon solutions and complexiton solutions to the Boussinesq equation. Various examples of exact solutions of those three kinds are computed. The newly presented Wronskian formulation is different from the one previously presented by Li et al., which does not yield rational solutions.

Third-order partial differential equations arising in the impulsive motion of a flat plate

We obtain numerical solutions to a class of third-order partial differential equations arising in the impulsive motion of a flat plate for various boundary data. In particular, we study the case of constant acceleration of the plate, the case of oscillation of the plate, and a case in which velocity is increasing yet acceleration is decreasing. We compare the numerical solutions with the known exact solutions in order to establish the validity of the method. Several figures illustrating both solution forms and the relative strength of the second and third-order terms are presented. The results obtained in this study reveal many interesting behaviors that warrant further study on the non-Newtonian fluid flow phenomena.

Conservation laws and conserved quantities for laminar two-dimensional and radial jets

A systematic way to derive the conserved quantities for the liquid jet, free jet and wall jet using conservation laws is presented. Both two-dimensional and radial jets are considered. The jet flows are described by Prandtl’s momentum boundary layer equation and the continuity equation. The multiplier approach (also know as variational derivative approach) is first applied to construct a basis of conserved vectors for the system. The basis consists of two conserved vectors. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived for the liquid jet and the free jet. The multiplier approach is then applied to construct a basis of conserved vectors for the third-order partial differential equation for the stream function. The basis consists of two local conserved vectors one of which is a non-local conserved vector for the system. The conserved quantities for the free jet and the wall jet are derived from the corresponding conservation laws and boundary conditions. The approach gives a unified treatment to the derivation of conserved quantities for jet flows and may lead to a new classification of jets through conserved vectors and their multipliers.

Physical Conserved Quantities for the Axisymmetric Liquid, Free and Wall Jets

A systematic way to derive the conserved quantities for the axisymmetric liquid jet, free jet and wall jet using conservation laws is presented. The flow in axisymmetric jets is governed by Prandtl’s momentum boundary layer equation and the continuity equation. The multiplier approach is used to construct a basis of conserved vectors for the system of two partial differential equations for the two velocity components. The basis consists of two conserved vectors. By integrating the corresponding conservation laws across the jet and imposing the boundary conditions, conserved quantities are derived for the axisymmetric liquid and free jet. The multiplier approach applied to the third-order partial differential equation for the stream function yields two local conserved vectors one of which is a non-local conserved vector for the system. One of the conserved vectors gives the conserved quantity for the axisymmetric free jet but the conserved quantity for the wall jet is not obtained from the second conserved vector. The conserved quantity for the axisymmetric wall jet is derived from a non-local conserved vector of the third-order partial differential equation for the stream function. This non-local conserved vector for the third-order partial differential equation for the stream function is obtained by using the stream function as multiplier. Keywords—Axisymmetric jet,liquid jet, free jet, wall jet, conservation laws, conserved quantity

Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays

We consider a problem of boundary feedback stabilization of first-order hyperbolic partial differential equations (PDEs). These equations serve as a model for physical phenomena such as traffic flows, chemical reactors, and heat exchangers. We design controllers using a backstepping method, which has been recently developed for parabolic PDEs. With the integral transformation and boundary feedback the unstable PDE is converted into a “delay line” system which converges to zero in finite time. We then apply this procedure to finite-dimensional systems with actuator and sensor delays to recover a well-known infinite-dimensional controller (analog of the Smith predictor for unstable plants). We also show that the proposed method can be used for the boundary control of a Korteweg–de Vries-like third-order PDE. The designs are illustrated with simulations.

Transient effects in oilfield cementing flows: Qualitative behaviour

We present an unsteady Hele–Shaw model of the fluid–fluid displacements that take place during primary cementing of an oil well, focusing on the case where one Herschel–Bulkley fluid displaces another along a long uniform section of the annulus. Such unsteady models consist of an advection equation for a fluid concentration field coupled to a third-order non-linear PDE (Partial differential equation) for the stream function, with a free boundary at the boundary of regions of stagnant fluid. These models, although complex, are necessary for the study of interfacial instability and the effects of flow pulsation, and remain considerably simpler and more efficient than computationally solving three-dimensional Navier–Stokes type models. Using methods from gradient flows, we demonstrate that our unsteady evolution equation for the stream function has a unique solution. The solution is continuous with respect to variations in the model physical data and will decay exponentially to a steady-state distribution if the data do not change with time. In the event that density differences between the fluids are small and that the fluids have a yield stress, then if the flow rate is decreased suddenly to zero, the stream function (hence velocity) decays to zero in a finite time. We verify these decay properties, using a numerical solution. We then use the numerical solution to study the effects of pulsating the flow rate on a typical displacement.

Initial-boundary value problems for linear PDEs with variable coefficients

AbstractA new approach for studying initial-boundary value problems for linear partial differential equations (PDEs) with variable coefficients was introduced recently by the second author, and was applied to PDEs involving second order derivatives. Here, we extend this approach further to solve an initial-boundary value problem for a third-order evolution PDE with a space-dependent coefficient. The analysis is presented in such a way that it can be applied to PDEs with higher derivatives, and thus provides a method for solving initial-boundary value problems for a certain class of linear evolution equations with variable coefficients of arbitrary order.

The integral bifurcation method and its application for solving a family of third-order dispersive PDEs

In this paper, an improved method named the integral bifurcation method is introduced. In order to demonstrate its effectiveness for obtaining travelling wave solutions of the nonlinear wave equations, a family of third-order dispersive partial differential equations which were given by A. Degasperis, D. Holm and A. Hone are studied. Many integral bifurcations are obtained for different parameter conditions. By using these integral bifurcations, many travelling wave solutions such as loop soliton solutions, solitary wave solutions, cusp soliton solutions and periodic wave solutions are obtained. In particular, under the conditions c 1 < 0 , c 2 = c 3 = 1 , a very peculiar periodic wave solution is obtained.

Series Solution for Unsteady Boundary-Layer Flows Due toImpulsively Stretching Plate

The third-order nonlinear partial differential equation modelling theunsteady boundary-layer flows caused by an impulsively stretchingflat plate is solved by using the Adomian decomposition method (ADM).The ADM yields analytic solution in the form of a rapidly convergentinfinite series with easily computable terms. The series solutionusing the ADM for the unsteady flows is presented for the first time.

On the existence of ion-acoustic solitary waves in a gravitating dusty plasma having charge fluctuation

Theoretical investigation on the propagation of ion-acoustic waves in an unmagnetized self-gravitating plasma has been made for the existence of solitary waves using the reductive perturbation method. It is observed that nonlinear excitations follow a coupled third-order partial differential equation which is slightly different from the usual case of coupled Korteweg-de Vries (K-dV) system. It appears that the system so deduced is a two-component generalization of the previous one derived by Paul et al. (1999) in which it was shown that ion-acoustic solitary waves can not exist in such system.

Generalization to higher dimensions of one-dimensional differential equation for Slater sum of an inhomogeneous electron liquid for a given local potential

The Slater sum (r, β) is the diagonal element of the canonical or Bloch density matrix, and by spatial integration yields the partition function. In one dimension, for independent electrons moving in a common potential v(x), the work of March and Murray (Phys. Rev., 120, 831 (1960)) already yielded a third-order partial differential equation for S(x, β). But to date, a generalization to higher dimensions for independent electrons in a given v(r) has not been effected. Here, using the differential virial equation (Holas and March, Phys. Rev. A, 51, 2040 (1995)) such generalization is derived. As a special case, the known differential equation for S(r, β) for three-dimensional spherically symmetric harmonic confinement is recovered. This equation is shown to be valid also for a wider, specific class of three-dimensional spherical systems.

Seismic wave propagating in Kelvin-Voigt homogeneous visco-elastic media

This paper studies, under a small disturbance, the responses of seismic transient wave in the visco-elastic media and the analytic solution of the corresponding third-order partial differential equation. A plane wave solution of Kelvin-Voigt homogeneous visco-elastic third-order partial differential equation with a pulse source is obtained. By the principle of pulse stacking of particle vibration, the result is extended to the solution of Kelvin-Voigt homogeneous visco-elastic third-order partial differential equation with any source. The velocities of seismic wave propagating and the attenuation of seismic wave in Kelvin-Voigt homogeneous visco-elastic media are discussed. The velocities of seismic wave propagating and the coefficient of attenuation of seismic wave in Kelvin-Voigt homogeneous visco-elastic media are derived, expressed as functions of density of the media, elastic modulus and visco-elastic coefficient. These results can be applied in inversing lithology parameters in geophysical prospecting.

On a Third-Order Partial Differential Equation

On a Third-Order Partial Differential Equation

The spectral representation of two-point boundary-value problems for third-order linear evolution partial differential equations

We use a spectral transform method to study general boundary-value problems for third-order, linear, evolution partial differential equations with constant coefficients, posed on a finite space domain. We show how this method yields a simple characterization of the discrete spectrum of the associated spatial differential operator, and discuss the obstructions that arise when trying to represent the solution of such a problem as a series of exponential functions. We first review the theory for second-order two-point boundary-value problems, and present an alternative way to derive the classical series representation, as well as an equivalent integral representation, which generally involves complex contours. We illustrate the advantages of the integral representation by studying in some detail the case where Robin-type boundary conditions are prescribed. We then consider the third-order case and show that the integral representation is in general not equivalent to a discrete series representation, justifying a posteriori the failure of some of the classical approaches. We illustrate the third-order case in detail, using the example of the equation q t + q xxx =0 for various types of boundary conditions. In contrast with the second-order case, the qualitative properties of the spectrum of the associated spatial differential operator depend in this case not only on the equation but also on the type of boundary conditions. In particular, the solution appears to admit a series representation only when the prescribed boundary conditions couple the two endpoints of the interval.