Approximate solutions to the conformable Rosenau-Hyman equation using the two-step Adomian decomposition method with Padé approximation

This paper adopts the Adomian decomposition method and the Padé approximation techniques to derive the approximate solutions of a conformable Rosenau-Hyman equation by considering the new definition of the Adomian polynomials. The Padé approximate solutions are derived along with interesting figures showing both the analytic and approximate solutions.     

Application of Lyapunov’s direct method to the study of the stability of solutions to systems of impulsive differential equations

Classical theorems on the stability of the solutions of impulsive differential equations are further developed.     

CTE method to the interaction solutions of Boussinesq–Burgers equations

Under investigation in this paper is the Boussinesq–Burgers equations, which describe the propagation of shallow water waves. Via the truncated Painlevé analysis and the consistent tanh expansion (CTE) method, some exact interaction solutions among different nonlinear excitations such as multiple resonant soliton solutions, soliton–error function waves, soliton–periodic waves, soliton–rational waves, and soliton–potential Burgers waves are explicitly given.     

Construction of asymptotic representations of solutions from the classL ρ τ for systems of differential equations by the method of characteristic matrix functions

By the method of characteristic matrix functions, we construct asymptotic representations of solutions of a system ofq linearn th-order differential equations with a singularity of rankp/r, p, r , in a sector of the complex plane whose central angle does not exceed r/p.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1148–1155, September, 1994.     

Bounded solutions of differential equations with “slow” time

The problem of existence of bounded solutions on the whole number line of a nonlinear system of first order with slow time is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1621–1623, November, 1992.     

A remark on the regularity of weak solutions to the Navier–Stokes equations in terms of the pressure in Lorentz spaces

For the incompressible Navier–Stokes equations in R³${\mathbb{R}}^3$, a regularity criterion for weak solutions is proved under the assumption that the pressure belongs to the scaling invariant Lorentz space with small norm, while corresponding results for the velocity field were proved by Sohr. The main theorem continues and extends a previous result given by the author.     

On a posteriori approximation of the set of solutions to a system of quadratic equations using Newton’s method

For quadratic systems of algebraic equations we propose an algorithm generating a posteriori estimates of the convex hull of the set of solutions using one step of Newton’s method. Results of some numerical tests are given.     

New soliton solutions of the generalized Zakharov equations using He’s variational approach

In this paper, we obtain new soliton solutions of the generalized Zakharov equations by the well-known He’s variational approach. The condition for continuation of the new solitary solution is obtained.     

Stability on a cone in terms of two measures for impulsive differential equations with “supremum”

The stability of nonlinear impulsive differential equations with “supremum” is studied. A special type of stability, combining two different measures and a dot product, is defined. The definition is a generalization of several types of stability known in the literature. Razumikhin’s method as well as a comparison method for scalar impulsive ordinary differential equations have been employed.     

Bounding the number of limit cycles of discontinuous differential systems by using Picard–Fuchs equations

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the upper bounds of the number of limit cycles given by the first order Melnikov function for discontinuous differential systems, which can bifurcate from the periodic orbits of quadratic reversible centers of genus one (r19): $\dot{x}=y?12x^2+16y^2$, $\dot{y}=?x?16xy$, and (r20): $\dot{x}=y+4x^2$, $\dot{y}=?x+16xy$, and the periodic orbits of the quadratic isochronous centers $(S_1):\dot{x}=?y+x^2?y^2$, $\dot{y}=x+2xy$, and $(S_2):\dot{x}=?y+x^2$, $\dot{y}=x+xy$. The systems (r19) and (r20) are perturbed inside the class of polynomial differential systems of degree n and the system $(S_1)$ and $(S_2)$ are perturbed inside the class of quadratic polynomial differential systems. The discontinuity is the line $y=0$. It is proved that the upper bounds of the number of limit cycles for systems (r19) and (r20) are respectively $4n?3(n\ge 4)$ and $4n+3(n\ge 3)$ counting the multiplicity, and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers $(S_1)$ and $(S_2)$ are exactly 5 and 6 (counting the multiplicity) on each period annulus respectively.     

Solvability of a system of equations for the Fourier–Chebyshev coefficients when solving ordinary differential equations by the Chebyshev series method

A solvability theorem for a system of equations with respect to approximate values of Fourier–Chebyshev coefficients is formulated. This theorem is a theoretical justification for numerical solution of ordinary differential equations using Chebyshev series.     

The “phase function” method to solve second-order asymptotically polynomial differential equations

The Liouville-Green (WKB) asymptotic theory is used along with the Bor?vka’s transformation theory, to obtain asymptotic approximations of “phase functions” for second-order linear differential equations, whose coefficients are asymptotically polynomial. An efficient numerical method to compute zeros of solutions or even the solutions themselves in such highly oscillatory problems is then derived. Numerical examples, where symbolic manipulations are also used, are provided to illustrate the performance of the method.     

Approximation of solutions of the Monge-Ampère equations by surfaces reduced to developable surfaces

We consider an approximate construction of the surface S that is the graph of a C ²-smooth solution z = z(x, y) of the parabolic Monge-Ampère equation

Reduction method for studying localized solutions of neural field equations on the Poincaré disk

We present a reduction method to study localized solutions of an integrodifferential equation defined on the Poincaré disk. This equation arises in a problem of texture perception modeling in the visual cortex. We first derive a partial differential equation which is equivalent to the initial integrodifferential equation and then deduce that localized solutions which are radially symmetric satisfy a fourth order ordinary differential equation.