Necessary and sufficient conditions for a generalized solution to the initial-boundary value problem for the wave equation to belong to W p 1 with p ≥ 1

We establish necessary and sufficient conditions on the boundary function under which a generalized solution to the initial-boundary value problem for the wave equation with boundary conditions of the first kind belongs to W _p ¹ .     

On some properties of generalized solutions of the wave equation in the classes L p and W p 1 for p ≥ 1

For each p ≥ 1, in closed analytic form, we establish the existence of a unique generalized solution in L _p of the mixed problem for the wave equation in the rectangle [0 ≤ x ≤ 1] × [0 ≤ t ≤ T] with zero initial conditions and with boundary conditions of the first kind, one of which is homogeneous. Next, we derive necessary conditions for this solution to belong to W _p ¹ . We present examples showing that these necessary conditions are not sufficient for any p ≥ 1.     

A necessary and sufficient condition for oscillation of the generalized Liénard equation

Summary In this paper we consider the Liénard system x= y – F(x), y= – g(x) and give a necessary and sufficient condition under which all solutions oscillate.     

A sufficient condition for the existence of a positive solution for a nonlinear fractional differential equation with the Riemann–Liouville derivative

In this work, by means of the fixed point theorem in a cone, we establish the existence result for a positive solution to a kind of boundary value problem for a nonlinear differential equation with a Riemann–Liouville fractional order derivative. An example illustrating our main result is given. Our results extend previous work in the area of boundary value problems of nonlinear fractional differential equations [C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett. 23 (2010) 1050–1055].     

Inverse problem for an equation of mixed type with the Lavrent’ev-Bitsadze operator and with a nonlocal boundary condition

For an equation of the mixed elliptic-hyperbolic type, we study the inverse problem with a nonlocal condition relating the derivatives of the solution on the elliptic and hyperbolic parts of the boundary. We prove a uniqueness criterion and construct the solution in the form of a Fourier series.     

Multiplicity of solutions to the Dirichlet problem for generalized Hénon equation

For the boundary value problem

On the solution of a generalized Schrödinger-type equation

Lu(t)+(u,F)g(t)=f(t), tS. , ( F, g). .

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The authors wish to thank Professor Yu. A. Rozanov for his help and discussions.     

The Cauchy problem for the wave equation with Lévy Laplacian

We present the solution of the Cauchy problem (the initial-value problem in the whole space) for the wave equation with infinite-dimensional Lévy Laplacian Δ _L , $$ \frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} = \Delta _L U(t,x) $$ in two function classes, the Shilov class and the Gâteaux class.     

Well-posedness of the poincar�� problem in a cylindrical domain for the higher-dimensional wave equation

It is known that waves (acoustic waves, radio waves, elastic waves, and electric waves) in cylindrical tubes are described by the wave equation. In the theory of hyperbolic-type partial differential equations, boundary-value problems with data on the whole boundary serve as examples of ill-posedness of the posed problems. In this work, it is shown that the Poincar´e problem in a cylindrical domain for the higher-dimensional wave equation is uniquely solvable. A uniqueness criterion for a regular solution is also obtained.     

Approximate solution to a singular,plane mixed boundary-value problem for Laplace’s equation in a curved rectangle

In this paper, we use a hybrid method based on a variant of Trefftz’s method (TM), in combination with the usual Boundary Collocation Method (BCM) to find the approximate solution to a singular, two-dimensional mixed boundary-value problem for Laplace’s equation in a rectangular sheet with one curved side.After expressing the solution as a finite linear combination of harmonic trial functions, the usual BCM is used to enforce the boundary condition on the curved side, while a variant of TM is applied to the three remaining sides. The singularity at one corner of the rectangle is treated via the enrichment of the expansion with a specially built harmonic function which has a singularity at one corner.The procedure ultimately produces a rectangular set of linear algebraic equations, which is solved by QR factorization method.Numerical results are presented and discussed, in order to assess the efficiency of the proposed method.     

On the behaviour ast→∞ of solutions of the second mixed problem for a second-order parabolic equation

In this paper is distinguished a geometric characteristic of the unbounded domain , that determines the rate of stabilization fort of the solution in (t>0)× of the second boundary value problem for a second-order parabolic equation, in which the initial function decreases sufficiently rapidly as |x|.