Efficient Modification of the Decomposition Method for Solving a System of PDEs

This paper presents an analysis solution for systems of partial differential equations using a new modification of the decomposition method to overcome the computational difficulties. Convergence of series solution was discussed with two illustrated examples, and the method showed a high-precision, being a fast approach to solve the non-linear system of PDEs with initial conditions. There is no need to convert the nonlinear terms into the linear ones due to the Adomian polynomials. The method does not require any discretization or assumption for a small parameter to be present in the problem. The steps of the suggested method are easily implemented, with high accuracy and rapid convergence to the exact solution, compared with other methods that can be used to solve systems of PDEs.

Strain-stress behavior of elastic layer in local loading

The paper studies the problem of a local loading of an elastic layer in 3D perspective. The solution of the boundary value problem subject to a concentrated force is constructed as a combination of two components. The first component is a classical solution of A. Lyav theory of elasticity, the second one is a solution proposed by I.М. Rapoport. The second component is distinctive in that it describes a point edge effect rapidly damping while moving off from the point of force application. This solution is built in a series form, namely, proper decompositions of the auxiliary nonself-adjoint differential operator. The convergence of series is ensured by a rapid growth of eigenvalues. Dying-away to zero at infinity is caused by the exponential law of Macdonald functions damping. The solution of the concentrated force action is used as a kernel to determine displacement vector components, tensors of deformations and strains in the problem of arbitrary local loading of an elastic layer. Eventually, analytical solution of the singular problem makes it possible to reasonably determine the strain-stress state in a local loading zone. Key words: strength, stress-strain behavior, theory of elasticity, differential equations, series, Bessel functions.

Schur Lemma and Uniform Convergence of Series through Convergence Methods

In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must satisfy in order to establish a result of Schur’s lemma type.

Thermosoluted Marangoni convective flow towards a permeable Riga surface

This study reveals the characteristics of chemical reaction on Marangoni mixed convective stream towards a penetrable Riga surface. The heat and mass phenomena are analysed within the sight of Dufour and Soret impacts. The administering partial differential equations system is converted into three nonlinear ordinary differential equations utilizing appropriately adjusted transformations. The resultant system of highly nonlinear equations is analytically solved by invoking the homotopy analysis method. Thereafter, the convergence of series solutions is discussed. The impact of appropriate parameters on various flow fields is thoroughly explained with the help of graphs and tables. The wall drag coefficient and relevant flux rates are arranged and discussed for dimensionless parameters. The outcomes show that the stronger Dufour effect of liquid causes a notable incremental variation in heat and mass flux, whereas an opposite trend is noted in the heat flux rate for the Soret effect. However, the mass flux is still found increasing for the stronger Soret effect.

Vortex Theory for Two Dimensional Boussinesq Equations

In this paper, the single center vortex method (SCVM) is extended to find some vortex solutions of finite core size for dissipative 2D Boussinesq equations. Solutions are expanded in to series of Hermite eigenfunctions. After confirmation the convergence of series of the solution, we show that, by considering the effect of temperature on the evolution of the vortex for the same initial condition as in [19] the symmetry of the vortex destroyed rapidly.

On the convergence of series of moments for row sums of random variables

Given a triangular array {Xn,k, 1 ? k ? n, n ? 1} of random variables satisfying E|Xn,k|p < ? for some p ? 1 and sequences {bn}, {cn} of positive real numbers, weshall prove that ??,n=1 cnE [|?n,k=1 (Xn,k-EXn,k)|/bn-?]p+ < ?, where x+ = max(x,0). Our results are announced in a general setting, allowing us to obtain the convergence of the series in question under various types of dependence.