Class Of Third-order Equations Research Articles

Applications of Solvable Lie Algebras to a Class of Third Order Equations

A family of third-order partial differential equations (PDEs) is analyzed. This family broadens out well-known PDEs such as the Korteweg-de Vries equation, the Gardner equation, and the Burgers equation, which model many real-world phenomena. Furthermore, several macroscopic models for semiconductors considering quantum effects—for example, models for the transmission of electrical lines and quantum hydrodynamic models—are governed by third-order PDEs of this family. For this family, all point symmetries have been derived. These symmetries are used to determine group-invariant solutions from three-dimensional solvable subgroups of the complete symmetry group, which allow us to reduce the given PDE to a first-order nonlinear ordinary differential equation (ODE). Finally, exact solutions are obtained by solving the first-order nonlinear ODEs or by taking into account the Type-II hidden symmetries that appear in the reduced second-order ODEs.

ОБ ОДНОЙ ЗАДАЧЕ ТИПА ДИРИХЛЕ ДЛЯ УРАВНЕНИЯ СОСТАВНОГО ТИПА

The aim of this paper is to prove the existence and uniqueness of smooth solutions to the Dirichlet type problem for one class of third-order equations that do not belong to any of the classic types. One of the main classes of non-classical equations is third-order composite type equations, the operator of which is a composition of first-order hyperbolic operator and an elliptic operator in the main part. A number of boundary value problems for the model composite type equations with the Laplace operator were investigated by T.D. Dzhuraev. Many studies have proved the existence of solutions to boundary value problems upon fulfillment of conditions of the convexity of area boundary. The method of proof used in this paper is similar to the method used in the research paper of the author mentioned above. For the research of composite type linear equations a combination of the method of potentials (Green's function) and integral identities is applied. The research method is based on reducing the studied problem with the help of the Green's function to an integral equation, the proof of its solubility and thus - the proof of the solvability of original problem. Upon fulfillment of certain conditions on given functions, a third-order equation reduces to a second-order equation of elliptic type with an unknown right-hand side and the boundary function. With the help of Green's functions for elliptic equations, the studied problem is reduced to a second-order equivalent integral equation, the solvability follows from Fredholm alternative and the theorem of uniqueness of the solution of the original problem

On the Agaciro Equation via the Scope of Green Function

We have undertaken an investigation of a kind of third-order equation called Agaciro equation within the folder of both integer and fractional order derivative. In the way of deriving the general exact solution of this equation, we employed the philosophy of the Green function together with some integral transform operators and special functions including but not limited to the Laplace, Fourier, and Mellin transform. We presented some examples of exact solution of this class of third-order equations for integer and fractional order derivative. It is important to point out that the value of Agaciro equation can be extended to describe assorted phenomenon in sciences.

‘Level grading’ a new graded algebra structure on differential polynomials: application to the classification of scalar evolution equations

We define a new grading, which we call the ‘level grading’, on the algebra of polynomials generated by the derivatives uk + i over the ring of C∞ functions of x, t, u, u1, …, uk, where . This grading has the property that the total derivative and the integration by parts with respect to x are filtered algebra maps. In addition, if u satisfies the evolution equation ut = F[u], where F is a polynomial of order m = k + p and of level p, then the total derivative with respect to t, Dt, is also a filtered algebra map. Furthermore, if the separant belongs to , then the canonical densities ρ(i) are polynomials of level 2i + 1 and Dtρ(i) is of level 2i + 1 + m. We define ‘KdV-like’ evolution equations as those equations for which all the odd canonical densities are non-trivial. We use the properties of level grading to obtain a preliminary classification of scalar evolution equations of orders m = 7, 9, 11, 13 up to their dependence on x, t, u, u1 and u2. These equations have the property that the canonical density ρ( − 1) is , where α, β and γ are functions of x, t, u, u1, u2. This form of ρ( − 1) is shared by the essentially nonlinear class of third order equations and a new class of fifth order equations.

Korteweg–de Vries and Kuramoto–Sivashinsky equations in bounded domains

where μ,ν are positive constants. This equation, in the case μ = 0, was derived independently by Sivashinsky [1] and Kuramoto [2] with the purpose to model amplitude and phase expansion of pattern formations in different physical situations, for example, in the theory of a flame propagation in turbulent flows of gaseous combustible mixtures, see Sivashinsky [1], and in the theory of turbulence of wave fronts in reaction–diffusion systems, Kuramoto [2]. The generalized KdV–KS equation (1.1) arises in modeling of long waves in a viscous fluid flowing down on an inclined plane. When ν = 0, we have the KdV equation studied by various authors [6–12]. From the mathematical point of a view, the history of the KdV equation is much longer than the one of the KS equation. Well-posedness of the Cauchy problem for the KdV equation in various classes of solutions was studied in [6–9]. Solvability of mixed problems for the KdV equation and for the KdV equation with dissipation in bounded domains studied Bubnov [11], Hublov [12], see also [19]. In [10], Bui An Ton proved well-posedness of the mixed problem for the KdV equation in (0,∞)× (0, T ) approximating the KdV equation by the KS type equations. Mixed problems for some classes of third order equations studied

Extension of some results concerning the generalized Liénard equation

A recent result of Mawhin [7] concerning the existence of forced oscillations for a second order equation of Lienard type with an arbitrary damping term is extended to some cases where the restoring force is not assumed to be sufficiently weak. The results are valid, too, for a certain class of third order equations. They are based on the Leray-Schauder principle. By an analogous argumentation the Rayleigh equation with a quasilinear restoring force, but with an arbitrary damping term, is shown to possess a periodic solution. Again, the result admits an extension to a third order equation.