On some properties of solutions to aP-type nonlinear equation

The Bäcklund transformation is constructed for a nonlinear ordinary differential equation (defining a polynomial Hamiltonian associated with the third Painlevé equation for γ = 0 and αδ ≠ 0). The nonlinear functional relationship is derived for solutions to this equation that correspond to different values of the parameter β entering it.     

On approximate solutions of the Pexider equation

Summary LetX be an abelian (topological) group andY a normed space. In this paper the following functional inequality is considered: {ie143-1} This inequality is a similar generalization of the Pexider equation as J. Tabor's generalization of the Cauchy equation (cf. [3], [4]). The solutions of our inequality have similar properties as the solutions of the Pexider equation. Continuity and related properties of the solutions are investigated as well.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On approximation of approximate solutions of Dhombres' equation

Let f:S→X map an abelian semigroup (S,+) into a Banach space (X‖⋅‖). We deal with stability of Dhombres' equation

     

On continuous approximate solutions of nonlinear differential equations

Two recently developed methods for continuous approximate solution of equations involving differential operators—the Adomian (decomposition) method and the Sarafyan method—are compared.     

Strong convergence of approximate solutions for nonlinear hyperbolic equation without convexity

Schonbek [M.E. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7 (1982) 959-1000] obtained the strong convergence of uniform bounded approximate solutions to hyperbolic scalar equation under the assumption that the flux function is strictly convex. While in this paper, by constructing four families of Lax entropies, we succeed in dealing with the non-convexity with the aid of the well-known Bernstein-Weierstrass theorem, and obtaining the strong convergence of uniform L^∞ or bounded viscosity solutions for scalar conservation law without convexity.     

On approximate solutions to some nonlinear diffusion problems

In the paper an approximate solution to a nonlinear diffusion problem in semiconductor production is considered. Some integral operator methods are used to estimate an error between this approximate solution and the exact one.This paper has been written when the author was a Visiting Professor in the Universidad de Extremadura.     

On existence and approximate solutions for a nonlinear integral equation arising in the theory of creep in metals

Existence of a solution to a nonlinear integral equation arising in the theory of creep in symmetric pressure vessels is established by means of a recursively defined sequence of functions. It is shown that this sequence converges monotonically and uniformly to the solution.

---

Zusammenfassung Das Problem des Kriechens in symmetrischen Druckgefässen führt auf eine nichtlineare Integralgleichung. Zum Beweis der Existenz einer Lösung dieser Gleichung wird rekursiv eine Folge von Näherungslösungen konstruiert. Es wird gezeigt, dass diese Folge monoton und gleichmässig nach der Lösung der Integralgleichung konvergiert.

---

---

This research was supported by the National Science Foundation under Grant MPS-75-07450.     

Invariant solutions of a nonlinear wave equation with a small dissipation obtained via approximate symmetries

Ricerche di Matematica - In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic...     

Qualitative analysis and approximate damped oscillatory solutions for a kind of nonlinear dispersive-dissipative equation

This paper makes qualitative analysis to the bounded traveling wave solutions for a kind of nonlinear dispersive-dissipative equation, and considers its solving problem. The relation is investigated between behavior of its solution and the dissipation coefficient. Further, all approximate damped oscillatory solutions when dissipation coefficient is small are presented by utilizing the method of undetermined coefficients according to the theory of rotated vector field in planar dynamical systems. Finally, error estimate is given by establishing the integral equation which reflects the relation between approximate and exact damped oscillatory solutions applying the idea of homogenization principle.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.     

On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation

The modified method of simplest equation is powerful tool for obtaining exact and approximate solutions of nonlinear PDEs. These solutions are constructed on the basis of solutions of more simple equations called simplest equations. In this paper we study the role of the simplest equation for the application of the modified method of simplest equation. We follow the idea that each function constructed as polynomial of a solution of a simplest equation is a solution of a class of nonlinear PDEs. We discuss three simplest equations: the equations of Bernoulli and Riccati and the elliptic equation. The applied algorithm is as follows. First a polynomial function is constructed on the basis of a simplest equation. Then we find nonlinear ODEs that have the constructed function as a particular solution. Finally we obtain nonlinear PDEs that by means of the traveling-wave ansatz can be reduced to the above ODEs. By means of this algorithm we make a first step towards identification of the above-mentioned classes of nonlinear PDEs.     

On the blow-up of solutions to a nonlinear dispersive rod equation

Using a variational approach we prove an optimal nonlinear convolution inequality. This result is then applied to give criteria for finite-time blow-up of solutions to a nonlinear model equation in elasticity, improving considerably upon recent blow-up results.     

On a method for constructing one-frequency solutions of a nonlinear wave equation

A method for constructing one-frequency solutions of nonlinear wave equations is suggested. This approach is based on a modified representation of asymptotic expansions by using special periodic Atebfunctions. This method makes it possible to obtain approximate solution of the problem under consideration without difficulty. L'viv Polytechnic Institute, L'viv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 782–784, June 1994.