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In this paper, the Kadometsev–Petviashvili equation and the Bargmann system are obtained from a second-order operator spectral problem L φ = (∂ 2 − v ∂ − λ u) φ = λ φ x . By means of the Euler–Lagrange equations, a suitable Jacobi–Ostrogradsky coordinate system is established. Using Cao's method and the associated Bargmann constraint, the Lax pairs of the differential equations are nonlinearized. Then, a new kind of finite-dimensional Hamilton system is generated. Moreover, involutive representations of the solutions of the Kadometsev–Petviashvili equation are derived. [ABSTRACT FROM AUTHOR]

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under consideration is reduced to a similar problem without the Bessel operator. The resulting auxiliary problem is solved by the Riemann method. On this basis, an exact solution of the original problem is constructed and analyzed. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation analogous to that of this matrix fourth Painlevé equation. Such auto-Bäcklund transformations, in appearance similar to those for Painlevé equations, are quite novel, having been little studied in the case of partial differential equations. Our work here shows the importance of the underlying structure of differential equations, whether ordinary or partial, in the derivation of such results. The starting point for the results in this paper is the construction of a new completely integrable equation, namely, an inverse matrix dispersive water wave equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider the differential equation y ″ + ω 2 ρ (x) y = 0 , where ω is a positive parameter. The principal concern here is to find conditions on the function ρ − 1 / 2 (x) which ensure that the consecutive differences of sequences constructed from the zeros of a nontrivial solution of the equation are regular in sign for sufficiently large ω. In particular, if c ν k (α) denotes the kth positive zero of the general Bessel (cylinder) function C ν (x ; α) = J ν (x) cos α − Y ν (x) sin α of order ν and if | ν | < 1 / 2 , we prove that (− 1) m Δ m + 2 c ν k (α) > 0 (m = 0 , 1 , 2 , ... ; k = 1 , 2 , ...) , where Δ a k = a k + 1 − a k. This type of inequalities was conjectured by Lorch and Szego in 1963. In addition, we show that the differences of the zeros of various orthogonal polynomials with higher degrees possess sign regularity. [ABSTRACT FROM AUTHOR]

In this work, the class of nonlinear complex fractional Kundu-Eckhaus equation is presented with a novel truncated M-fractional derivative. This model is significant and notable in quantum mechanics with good-natured physical characteristics. The motivation for this paper is to construct new solitary and kink wave solutions for the governing equation using the ansatz method. A complex-fractional transformation is applied to convert the fractional Kundu-Eckhaus equation into an ordinary differential equations system. The equilibria of the corresponding dynamical system will be presented to show the existence of traveling wave solutions for the governing model. A novel kink and solitary wave solutions of the governing model are realized by means of the proposed method. In order to gain insight into the underlying dynamics of the obtained solutions, some graphical representations are drawn. For more illustration, several numerical applications are given and analyzed graphically to demonstrate the ability and reliability of the method in dealing with various fractional engineering and physical problems. [ABSTRACT FROM AUTHOR]

In this paper, we adopt the choice theory of the shape parameters contained in the smooth radial basis functions to solve Poisson equations. Luh's choice theory, based on harmonic analysis, is mathematically complicated and applies only to function interpolation. Here, we aim at presenting an easily accessible approach to solving differential equations with the choice theory which proves to be very successful, not only by its easy accessibility but also by its striking accuracy and efficiency. Our emphases are on the highly reliable prediction of the optimal value of the shape parameter and the extremely small approximation errors of the numerical solutions to the differential equations. We hope that our approach can be accepted by both mathematicians and non-mathematicians. [ABSTRACT FROM AUTHOR]

The main objective of this paper is concerned with the convergence of the boundary parameter for the large-scale, three-dimensional, viscous primitive equations. Such equations are often used for weather prediction and climate change. Under the assumptions of some boundary conditions, we obtain a prior bounds for the solutions of the equations by using the differential inequality technology and method of the energy estimates, and the convergence of the equations on the boundary parameter is proved. [ABSTRACT FROM AUTHOR]

In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics − c 2 + ∥ x ∥ 2 when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments demonstrate that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c. [ABSTRACT FROM AUTHOR]

The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution's blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for differential–difference parabolic equations with potentials undergoing translations with respect to the spatial-independent variable. Such equations are used for the modeling of various phenomena not covered by the classical theory of differential equations (such as nonlinear optics, nonclassical diffusion, multilayer plates and envelopes, and others). From the viewpoint of the pure theory, they are important due to crucially new effects not arising in the case of differential equations and due to the fact that a number of classical methods, tools, and approaches turn out to be inapplicable in the nonlocal theory. The qualitative novelty of our investigation is that the initial-value function is assumed to be summable. Earlier, only the case of bounded (essentially bounded) initial-value functions was investigated. For the prototype problem (the spatial variable is single and the nonlocal term of the equation is single), we construct the integral representation of a solution and show its smoothness in the open half-plane. Further, we find a condition binding the coefficient at the nonlocal potential and the length of its translation such that this condition guarantees the uniform decay (weighted decay) of the constructed solution under the unbounded growth of time. The rate of this decay (weighted decay) is estimated as well. [ABSTRACT FROM AUTHOR]

In this study, we investigate coupled real and complex dispersionless equations for curve families, even if they have singular points. Even though the connections with the differential equations and regular curves were considered in various ways in the past, since each curve does not need to be regular, we establish the connections for framed base curves, which generalize regular curves with linear independent conditions. Also, we give the Lax pairs of the real and complex coupled dispersionless equations from the motions of any framed curve. These give us significant conditions based on the framed curvatures and associated curvatures of the framed curves for integrability since it is well known that the Lax pair provides the integrability of differential equations. [ABSTRACT FROM AUTHOR]

We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation. [ABSTRACT FROM AUTHOR]

For generalized ordinary differential equations, sufficient criteria are given for the Lipschitz stability in terms of two measures of the trivial solutions. As an application, we apply our main results by studying the Lipschitz stability for measure differential equations and impulsive differential equations. Compared to the classical ones, the conditions here regarding the functions are more general. [ABSTRACT FROM AUTHOR]

In this paper we present a two parameter family of differential equations treated by Jacopo Riccati, which does not appear in any modern repertoires and we extend the original solution method to a four parameter family of equations, translating the Riccati approach in terms of Lie symmetries. To get the complete solution, hypergeometric functions come into play, which, of course, were unknown in Riccati's time. Re-discovering the method introduced by Riccati, called by himself dimidiata separazione (splitted separation), we arrive at the closed form integration of a differential equation, more general to the one treated in Riccati's contribution, and which also does not appear in the known repertoires. [ABSTRACT FROM AUTHOR]

In this paper, we present further developed results on Hille–Wintner-type integral comparison theorems for second-order half-linear differential equations. Compared equations are seen as perturbations of a given non-oscillatory equation, which allows studying the equations on the borderline of oscillation and non-oscillation. We bring a new comparison theorem and apply it to the so-called generalized Riemann–Weber equation (also referred to as a Euler-type equation). [ABSTRACT FROM AUTHOR]

This research work is dedicated to solving the n-generalized Korteweg–de Vries (KdV) equation in a fractional sense. The method is a combination of the Sumudu transform and the Adomian decomposition method. This method has significant advantages for solving differential equations that are both linear and nonlinear. It is easy to find the solutions to fractional-order PDEs with less computing labor. [ABSTRACT FROM AUTHOR]

This paper aims to study the oscillatory properties of fourth-order advanced differential equations with p-Laplacian like operator. By using the technique of Riccati transformation and the theory of comparison with first-order delay equations, we will establish some new oscillation criteria for this equation. Some examples are considered to illustrate the main results. [ABSTRACT FROM AUTHOR]

Iterative methods were employed to obtain solutions of linear and non-linear systems of equations, solutions of differential equations, and roots of equations. In this paper, it was proved that s-iteration with error and Picard–Mann iteration with error converge strongly to the unique fixed point of Lipschitzian strongly pseudo-contractive mapping. This convergence was almost F -stable and F -stable. Applications of these results have been given to the operator equations F x = f and x + F x = f , where F is a strongly accretive and accretive mappings of X into itself. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations arising from the generating function of the (r , β) -Bell polynomials. We give explicit identities for the (r , β) -Bell polynomials. Finally, we find the zeros of the (r , β) -Bell equations with numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we consider conformal mappings of Riemannian spaces onto Ricci-2-symmetric Riemannian spaces and geodesic mappings of spaces with affine connections onto Ricci-2-symmetric spaces. The main equations for the mappings are obtained as a closed system of Cauchy-type differential equations in covariant derivatives. We find the number of essential parameters which the solution of the system depends on. A similar approach was applied for the case of conformal mappings of Riemannian spaces onto Ricci-m-symmetric Riemannian spaces, as well as geodesic mappings of spaces with affine connections onto Ricci-m-symmetric spaces. [ABSTRACT FROM AUTHOR]

In this work, we develop enhanced Hille-type oscillation conditions for arbitrary-time, second-order quasilinear functional dynamic equations. These findings extend and improve previous research that has been published in the literature. Some examples are given to demonstrate the importance of the obtained results. [ABSTRACT FROM AUTHOR]

The existence, uniqueness and uniformly L p estimates for solutions of a high-order abstract Navier–Stokes problem on half space are derived. The equation involves an abstract operator in a Banach space E and small parameters. Since the Banach space E is arbitrary and A is a possible linear operator, by choosing spaces E and operators A, the existence, uniqueness and L p estimates of solutions for numerous classes of Navier–Stokes type problems are obtained. In application, the existence, uniqueness and uniformly L p estimates for the solution of the Wentzell–Robin-type mixed problem for the Navier–Stokes equation and mixed problem for degenerate Navier–Stokes equations are established. [ABSTRACT FROM AUTHOR]