General Integro-differential Equations Research Articles

The Parabolic Transform and Some Singular Integral Evolution Equations

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.

A New Regularization Mechanism for the Boltzmann Equation Without Cut-Off

We apply recent results on regularity for general integro-differential equations to derive a priori estimates in Hölder spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in $${L^\infty}$$ which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded.

Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

Abstract We consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of the Riemann-Liouville integrals, the equations we analyze can be understood as equations with time-varying coefficients. Replacing the Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchy-type problems can be solved numerically, and, in some cases analytically, in terms of the Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.

Existence Results on General Integrodifferential Evolution Equations in Banach Space

In this paper we prove the existence of mild solutions of a general class of nonlinear evolution integrodifferential equation in Banach spaces. Based on the resolvent operator and the Schaefer fixed point theorem, a sufficient condition for the existence of general integrodifferential evolution equations is established.

Spectral Method for Solving The General Form Linear Fredholm-Volterra Integro Differential Equations Based on Chebyshev Polynomials

The main aim of this paper reports a pseudo spectral method based on integrated Chebyshev polynomials for numerically solving higher order linear Fredholm Volterra integro differential equations (FVIDE) in the most general form. This method transforms FVIDE to matrix equations which correspond to a system of linear algebraic equations with unknown Chebyshev coefficients. The present numerical results are in satisfactory agreement with the exact solutions and show the advantages of this method to some other known methods.

General integro-differential equations and optimal controls on Banach spaces

In this paper a general integro-differential equations on Banach space are considered. Existence of $\alpha $-mild solutions is proved. Existence of optimal controls of systems governed by general integro-differential equations is also presented.

The thickness functions of the shells of revolution subjected to axisymmetrical loads

The inverse problems for determining the meridian shape or varying thickness function of momentless shells of revolution under given loads were concerned in many works [2, 3, 4]. However, for the complexity of loads or configuration of a shell these problems haven' t bee.n solved perfectly because of its mathematical difficulties. In this paper, the problem for determining the thickness function of shells of revolution such as a parabola, sphere arc! under axisymmetrical loads is considered. The general integro-differential equations for determination of the meridian form and shell thickness are obtained. A solution of differential equations by semi-analytical and numerical methods for the thickness is presented. The numerical solutions are given for the parabola under external pressure, the sphere immerged in the fluid and the sphere arc. Obtained results may be used in the thin shell design.

Ginzburg-Landau equations for anisotropic superconductors

We use the framework of a general quasiclassical theory of superconductivity which allows for arbitrary gap and Fermi surface anisotropy and for impurity scattering in Born approximation. We derive general Ginzburg-Landau integro-differential equations, which comprise all previous limiting cases considered in the literature. From these equations more specialized Ginzburg-Landau equations may easily be derived.

Extended backward differentiation formulae in the numerical solution of general Volterra integro-differential equations

In this paper nonlinear Volterra integro-differential equations are considered with kernels of the formP(x,s,y(s)) andK(x,s,y(x),y(s)) and extended backward differentiation methods are applied as extended from their introduction for the solution of ordinary differential equations by Cash [4]. An error bound is obtained and a rate of convergence is found and validated by testing the method on some examples. The numerical results are compared with those obtained by applying standard backward differentiation and collocation methods.

Oscillations of solutions of general integrodifferential equations

Oscillations of solutions of general integrodifferential equations

On Choice of Trial Functions in Integro-Differential Variational Principles of Transport Theory

In several problems of particle transport, quantities of macroscopic interest can be related to stationary values of variational functionals based on general integro-differential equations and boundary conditions. Within the context of the jump (Milne's) problem, it is shown how highly accurate results can be obtained by using trial functions based on the eigenfunctions of the relevant integrodifferential equations. Such choices of trial functions should apply equally effectively to problems in curved geometries, both internal and external.

A self-consistent theory of steady-state lamellar solidification in binary eutectic systems

The potential theoretic methods developed recently at NRL for solving the diffusion equation are applied to the free-boundary problem which describes lamellar solidification in binary eutectic systems. By using these techniques, the original free-boundary problem is reduced to a set of coupled nonlinear integro-differential equations which when solved yield the shape of the solid/liquid interface and the solute concentration on the interface. A simplified version of the general integro-differential equations is derived by assuming that (1) the solute diffusion length is large compared with the lamellar spacing, and (2) the solid/liquid interface is approximately isothermal. Numerical solutions to the approximate equations are obtained and are utilized to assess the effect of interface curvature on the interfacial solute concentrations, and to check the new theory for consistency with experiment.