Solutions Of Nonhomogeneous Equations Research Articles

Mellin Transform and Integro-Differential Equations with Logarithmic Singularity in the Kernel

In the one-dimensional domain it is studied the unique solvability of a class of integro-differential equations with logarithmic singularity in the kernels. The solution of the considering integro-differential equations with logarithmic singularity in the kernels on dependent of roots of corresponding characteristic equations is obtained in an explicit form. The first part of paper deals with the introduction of the Mellin transformation as a holomorphic function in the complex plane. The definition of this integral transform is given and its main properties are described in details. The second part deals with the application of the Mellin transform to the solution of one-dimensional integro-differential equations with the logarithmic singularity in the kernels. Here, in order to obtain particular solution of nonhomogeneous equations, we applied the analogies of Duhamel’s theorem. It is shown that, if the kernel of studied equation has first order logarithmic singularity of polynomial’s form, then to this equation corresponds characteristic (algebraic) equation of the third order. In this case the solution is found by means of degree of logarithmic and trigonometric functions. If the integro-differential equation has $$n$$ -th order logarithmic singularity of polynomial’s form, then to this equation corresponds $$n$$ -th order characteristic equation. At three main cases the solution is found in an explicit form. Also, it is established that the degree of logarithmic singularity acts to number of linearly independent solutions of given classes of integro-differential equations.

A time-domain symplectic method for finite viscoelastic cylinders

The symplectic method is introduced for boundary-condition problems of finite viscoelastic cylinders. On the basis of the state space formalism and the use of the Laplace integral transform, the general solution of the governing equations, zero- and nonzero-eigenvalue eigenvectors, are obtained. Since the eigenvectors are expressed in concise analytical forms, the adjoint symplectic relation of the Laplace domain is generalized to the time domain. Therefore, the particular solution and the eigenvector expansion method can be discussed directly in the eigenvector space of the time domain, without employing the iterative application of the inverse Laplace transformation. Using this method, various boundary conditions, the particular solution of nonhomogeneous equations, especially the interfacial continuity conditions of composite materials, can be conveniently described by combinations of the eigenvectors.

Sturm theorems for second order linear nonhomogenous differential equations and localization of zeros of the solution

It is shown that Sturm theorems, formulated in the 1830’s ([1], [2], [3] and [4]) and valid for second order linear homogeneous differential equation L(y)≡y″+a(x)y′+b(x)y=0, could as well be formulated for the class of nonhomogeneous linear differential equations L(y)=f(x). Criteria for the existence of oscillatory solutions of nonhomogeneous equations, as well as more exact locations of the zeros are given.

Rate of convergence of the method of fundamental solutions and hyperinterpolation for modified Helmholtz equations on the unit ball

Approximate solutions of boundary value problems of homogeneous modified Helmholtz equations on the unit ball are explicitly constructed by the method of fundamental solutions (MFS) with the order of approximation provided. Hyperinterpolation is used to find particular solutions of non-homogeneous equations, and the rate of convergence of solving boundary value problems of non-homogeneous equations is derived. Numerical examples are shown to demonstrate the efficiency of the methods.

An index classification theory of homogeneous p-Laplacian equations and existence of solutions of non-homogeneous equations

An index classification theory of homogeneous p-Laplacian equations and existence of solutions of non-homogeneous equations

Evolutionary Semigroups and Dichotomy of Linear Skew-Product Flows on Locally Compact Spaces with Banach Fibers

We study evolutionary semigroups generated by a strongly continuous semi-cocycle over a locally compact metric space acting on Banach fibers. This setting simultaneously covers evolutionary semigroups arising from non-autonomous abstract Cauchy problems andC0-semigroups, and linear skew-product flows. The spectral mapping theorem for these semigroups is proved. The hyperbolicity of the semigroup is related to the exponential dichotomy of the corresponding linear skew-product flow. To this end a Banach algebra of weighted composition operators is studied. The results are applied in the study of: “roughness” of the dichotomy, dichotomy and solutions of nonhomogeneous equations, Green's function for a linear skew-product flow, “pointwise” dichotomy versus “global” dichotomy, and evolutionary semigroups along trajectories of the flow.

Nonoscillation of solutions of a class of third order differential equations

wherep, q and a are real-valued continuous functions on [0, ~) such that e (s )>0, have been the subject of investigation for many authors. The general properties of solutions of a class of linear homogeneous third order differential equations were first studied by Birkhoff [2]. Further results were obtained by Gregus [4]. Hanan [5] studied oscillatory and nonoscillatory behaviour of solutions of (1.1) with a continuously differentiable and a ( s ) l . Lazer[6] studied oscillatory behaviour of solutions of (1.1) with a ( s )= 1. The author in [7] and S. Parhi and the author in [8, 9, 10] obtained sufficient conditions for oscillation and nonoscillation of solutions of nonhomogeneous equations