[en] In this paper, a semi-discrete scheme is presented for solving fourth-order partial integro-differential equations with a weakly singular kernel. The second-order backward difference formula is used to discretize the temporal derivatives. After discretizing the temporal derivatives, the considered problems are converted into a set of ordinary differential equations which are solved by using Legendre wavelets collocation method. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Several numerical examples are included to demonstrate the accuracy and efficiency of the proposed method.

[en] In this paper, we first define generalized shifted Jacobi polynomial on interval and then use it to define Jacobi wavelet. Then, the operational matrix of fractional integration for Jacobi wavelet is being derived to solve fractional differential equation and fractional integro-differential equation. This method can be seen as a generalization of other orthogonal wavelet operational methods, e.g. Legendre wavelets, Chebyshev wavelets of 1st kind, Chebyshev wavelets of 2nd kind, etc. which are special cases of the Jacobi wavelets. We apply our method to a special type of fractional integro-differential equation of Fredholm type. (paper)

[en] Various cases where the construction of exact solutions of nonlinear equations of mathematical physics leads to overdetermined systems of ordinary differential equations with parameters that are not included in the original partial differential equation are described. An appropriate nonclassical problem for ordinary differential equations with parameters is formulated and the new concept of the conditional cost of the exact solution is introduced

[en] Nonlinear partial differential equations of parabolic and hyperbolic types with delay have been considered in the paper. A number of exact traveling-wave solutions and some solutions of other kinds have been also described

[en] A new approach to the use of the Lie group technique for partial and ordinary differential equations dependent on a small parameter is developed. In addition to determining approximate solutions to the perturbed equation, the approach allows constructing integrable equations that have solutions with (partially) prescribed features. Examples of application of the approach to partial differential equations are given

[en] In this paper, the sufficient conditions for the existence of periodic solutions of the certain fifth order differential equations are given. 6 refs

[en] In this paper we study the nonoscillatory behaviour of even-order functional differential inequalities and equations with continuous distributed retarded and advanced arguments. For arguments we give some conditions under which these inequalities only have nonoscillatory solutions of degree 0 or n. (author)

No abstract available

[en] If the initial and boundary data for a PDE do not obey an infinite set of compatibility conditions, singularities will arise in the solution at the corners of the initial time-space domain. For dissipative equations, such as the 1-D heat equation or 1-D convection-diffusion equations, the impacts of these singularities are short lived. However, they can cause a very severe loss of numerical accuracy if we are interested in transient solutions. The phenomenon has been described earlier from a theoretical standpoint. Here, we illustrate it graphically and present a simple remedy which, with only little extra cost and effort, restores full numerical accuracy

[en] This book is a compendium of thermodynamic relationships for pure substances. In chapters 1 and 2, the basic mathematical background and general forms of the First and Second Laws are presented. Chapter 3 introduces the four thermodynamic potentials, together with methods for calculating other thermodynamic quantities. Alternate potentials are discussed briefly, but with enough detail to allow a thorough understanding of the possible applications. The Maxwell relations are introduced in chapter 4, both for generalized forces and variable amounts of substance. Chapter 5 covers applications to simple systems. The Gibbs-Helmholtz equations, specific heat calculations and flow systems, subsonic and supersonic, are covered. The possibilities of accelerating supersonic or subsonic flows by means of work, heat exchange, or frictional dissipation are presented by means of elegant mathematical demonstrations, that flow easily from the general relationships derived in the previous chapter. Chapter 6 is wholly devoted to the occurrence and interpretation of discontinuities taking place at phase boundaries, and chapter 7 deals with phase transitions. The critical point and systems under the action of different fields, as well as surface tension effects are discussed in the last two chapters