Method For Integro-differential Equations Research Articles

A Hybrid Stress Finite Element Method for Integro-Differential Equations Modelling Dynamic Fractional Order Viscoelasticity

A Hybrid Stress Finite Element Method for Integro-Differential Equations Modelling Dynamic Fractional Order Viscoelasticity

Mixed virtual element method for integro-differential equations of parabolic type

Mixed virtual element method for integro-differential equations of parabolic type

Determination of the scope of the experimental-calculation method for measuring the touch voltage

The work is devoted to the improvement of methods for determining the normalized parameters of the grounding system (GS) of operating power stations and substations. The aim of the work is determination of the scope of the experimental-calculated method for measuring the touch voltage, depending on short-circuit (SC) current value for the given dimensions of the GS and the type of soil. Methodology. The study analyzed the non-linear effect of the SC current value on the touch voltage, taking into account such factors as the GS size and the soil type. The calculation used statistical data on the GS size and the characteristics of the soil obtained by monitoring the GS state of 585 operating electrical substations with a voltage class of 110-750 kV using the induction method and the method of vertical soil sounding, respectively. For the calculation, a mathematical model of a non-equipotential GS located in a three-layer semiconductor space with plane-parallel boundaries was used, this model was developed using the method of integro-differential equations. Results. To determine the scope of the method, in this work it is proposed to use the linearity criterion, which is determined due to the ratio of the constant of reduced touch voltage to the current value. The example shows the method for determining the threshold minimum and maximum values of the measuring current of the soil, in the range between which the measurements by experimental-calculated methods are impossible. A table of threshold current values has been formed and recommendations have been developed on the possibility of using experimental-measuring methods for determining the touch voltage depending on the GS size and soil characteristics.

A New Decomposition Method for Integro-Differential Equations

This present study developed a new Modified Adomian Decomposition Method (MADM) for integro-differential equations. The modification was carried out by decomposing the source term function into series. The terms in the series were then selected in pairs to form the initials for the prevailing approximation. The newly modified Adomian decomposition method (MADM) accelerates the convergence of the solution faster than the Standard Adomian Decomposition Method (SADM). This study recommends the use of the MADM for solving integro-differential equations.

Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

The aim of this paper is to carry out an improved analysis of the convergence of the Nyström and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree ⩽r−1, we obtain convergence order 2r for degenerate kernel and Nyström methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r+1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.

Efficient decomposition method for integro-differential equations

Different methods have been used in the solution of integro-differential equations. Many of these methods such as Standard Adomian Decomposition Method (SADM) take several iterations which might be difficult to solve and also consume time before getting an approximation. This present study developed a new Modified Adomian Decomposition Method (MADM) for Integro-Differential Equations. The modification was carried out by decomposing the source term function into series. The newly modified Adomian decomposition method (MADM) accelerates the convergence of the solution (MADM) faster the Standard Adomian Decomposition Method (SADM). This study recommends the use of MADM for solving Integro-Differential Equations.

Method of integro-differential equations for interpreting the results of vertical electrical sounding of the soil

The paper is devoted to the problem of determining the geoelectric structure of the soil within the procedure of testing the grounding arrangements of existing power plants and substations to the required depth in conditions of dense development. To solve the problem, it was proposed to use the Schlumbergers method , which has a greater sounding depth compared to the Wenner electrode array. The purpose of the work is to develop a mathematical model for interpreting the results of soil sounding by the Schlumberger method in the form of a four-layer geoelectric structure. Methodology. To construct a mathematical model, it is proposed to use the solution of a particular problem about the field of a point current source, which, like the observation point, is located in the first layer of a four-layer soil. Based on this expressions, a system of linear algebraic equations of the 7-th order with respect to the unknown coefficients ai and bi was compiled. On the basis of its analytical solution, an expression for the potential of the electric field was obtained for conducting VES (the point current source and the observation point are located only on the soil surface). Results. Comparison of the results of soil sounding by the Schlumberger installation and the interpretation of its results for the same points shows a sufficient degree of approximation: the maximum relative error does not exceed 9.7 % (for the second point), and the average relative error is 3.6 %. Originality. Based on the obtained expression, a test version of the program was implemented in Visual Basic for Applications to interpret the results of VES by the Schlumberger method. To check the obtained expressions, the interpretation of the VES results was carried out on the territory of a 150 kV substation of one of the mining and processing plants in the city of Kriviy Rih. Practical significance. The developed mathematical model will make it possible to increase the sounding depth, and, consequently, the accuracy of determining the standardized parameters of the grounding arrangements of power stations and substations.

Investigation of the evaporation in the boundary layer

THE PURPOSE. Consider a stationary diffusion problem when a pure liquid evaporates from a flat surface of evaporation into a laminar boundary layer of a forced gas flow (in the absence of deepening of the evaporation surface and wave formation on it) at the number. In the classical model of the diffusion problem of the flow of mass from a flat surface into the laminar boundary layer, only the additional slowing down effect arising in this case is taken into account. However, the resulting solution does not correspond to the general case of evaporation, since in this case the mass transfer can significantly depend on the thermal conditions of the problem, conjugate in phases; in criterion form, this circumstance is expressed by the appearance of an additional parameter [1-3]. Note that this parameter is related to the value of the derivative of the relative concentration along the transverse coordinate on the evaporation surface. In the course of the proposed solution, the temperature of the evaporation surface and, accordingly, the value of this parameter were taken constant. METHODS. When solving the problem, we used approximate numerical methods for integrating the diffusion equation (Euler's method, integro-differential equation method, and also the method of successive approximations). In this case, the retarding effect of the vapor flow from the surface of the phase transition was assumed to be relatively insignificant in our case (which corresponded to the experimental data used in [1-3]. RESULTS. The article analyzes the well-known classical solution of the diffusion equation according to the Hartnett - Eckert model and notes that the result obtained in this case does not correspond to the general case of evaporation, when the mass transfer in the gas phase also depends on the complex. Based on the solution obtained in our work, we come to the conclusion that the effect of this parameter manifests itself in an increase in the thickness of the diffusion boundary layer. In addition, this effect is also associated with the value of the longitudinal coordinate, being more noticeable at its small values. CONCLUSION. The indicated evaporation pattern can be physically explained by a relatively larger amount of evaporated substance than in the “standard” case (since values, in turn, are associated with higher values of the evaporation surface temperature). It can also be assumed that in the region of the gas flow immediately adjacent to the evaporation surface, these factors manifest themselves in a similar way in the case of turbulent flows.

Quadrature methods for integro-differential equations of Prandtl’s type in weighted spaces of continuous functions

The paper deals with the approximate solution of integro-differential equations of Prandtl’s type. Quadrature methods involving “optimal” Lagrange interpolation processes are proposed and conditions under which they are stable and convergent in suitable weighted spaces of continuous functions are proved.The efficiency of the method has been tested by some numerical experiments, some of them including comparisons with other numerical procedures. In particular, as an application, we have implemented the method for solving Prandtl’s equation governing the circulation air flow along the contour of a plane wing profile, in the case of elliptic or rectangular wing-shape.

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

This paper presents analytical higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system based on an integro-differential equation method for obtaining periodic solutions to nonlinear differential equations. The stability of the limit cycles obtained was then investigated using a method for carrying out Floquet analysis based on developments to extensions of the method for solving Hill’s Determinant arising in analysing the Mathieu equation, which have previously been reported in the literature. The results of the Floquet analysis, together with the limit cycle predictions, have then been used to provide some estimates of points on the boundary of the domain of attraction of stable equilibrium points arising from a sub-critical Hopf bifurcation. This was achieved by producing a local approximation to the stable manifold of the unstable limit cycle that occurs.The integro-differential equation to be solved for the limit cycles involves no approximations. These only arise through the iterative approach adopted for its solution in which the first approximation is that which would be obtained from the harmonic balance method using only fundamental frequency terms. The higher order approximations are shown to give significantly improved predictions for the limit cycles for the cases considered. The Floquet analysis based approach to predicting the boundary of domains of attraction met with some success for conditions just following a sub-critical Hopf bifurcation.Although this study has focussed on cubic non-linearities, the method presented here could equally be used to refine limit cycle predictions for other non-linearity types.

EXTENSION AND DECOMPOSITION METHOD FOR DIFFERENTIAL AND INTEGRO-DIFFERENTIAL EQUATIONS

EXTENSION AND DECOMPOSITION METHOD FOR DIFFERENTIAL AND INTEGRO-DIFFERENTIAL EQUATIONS

Generalized Spline Method for Integro-Differential Equations of Fractional Order

In This paper generalized spline method and Caputo differential operator is applied to solve linear fractional integro-differential equations of the second kind. Comparison of the applied method with exact solutions reveals that the method is tremendously effective.

A solution method for integro-differential equations of conformable fractional derivative

The aim of this work is to determine an approximate solution of a fractional order Volterra-Fredholm integro-differential equation using by the Sinc-collocation method. Conformable derivative is considered for the fractional derivatives. Some numerical examples having exact solutions are approximately solved. The comparisons of the exact and the approximate solutions of the examples are presented both in tables and graphical forms.

Dynamical density functional theory with hydrodynamic interactions in confined geometries.

We study the dynamics of colloidal fluids in both unconfined geometries and when confined by a hard wall. Under minimal assumptions, we derive a dynamical density functional theory (DDFT) which includes hydrodynamic interactions (HI; bath-mediated forces). By using an efficient numerical scheme based on pseudospectral methods for integro-differential equations, we demonstrate its excellent agreement with the full underlying Langevin equations for systems of hard disks in partial confinement. We further use the derived DDFT formalism to elucidate the crucial effects of HI in confined systems.

Order-optimal methods for integro-differential equations in the singular case

We study a linear integro-differential equation with a coefficient that has zeros of finite orders. For its approximate solution in the space of generalized functions, we suggest and justify special generalized versions of spline methods. Their optimization in the accuracy order is carried out.

Some Remarks on the Duality Method for Integro-Differential Equations with Measure Data

Abstract We deal with existence, uniqueness and regularity for solutions of the boundary value problem { ℒ s ⁢ u = μ in ⁢ Ω , u ⁢ ( x ) = 0 on ⁢ ℝ n \ Ω , $\left\{\begin{aligned} \displaystyle\mathcal{L}^{s}u&\displaystyle=\mu&&% \displaystyle\text{in }\Omega,\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\mathbb{R}^{n}% \backslash\Omega,\end{aligned}\right.$ where Ω is a bounded domain of ℝ n ${\mathbb{R}^{n}}$ , μ is a bounded Radon measure on Ω, and ℒ s ${\mathcal{L}^{s}}$ is a non-local operator of fractional order s whose kernel K is comparable with the one of the fractional Laplacian.

Displacement discontinuity analysis of a nonlinear interfacial crack in three-dimensional transversely isotropic magneto-electro-elastic bi-materials

The displacement discontinuity method is introduced and extended to study the electric and magnetic nonlinear effect of an interfacial crack in three-dimensional magneto-electro-elastic bi-materials under combined loadings. Green’s functions due to uniformly distributed extended displacement discontinuities over a ring element are derived via the extended displacement discontinuity integro-differential equation method. The electric−magnetic polarization saturation model is adopted for the electric and magnetic nonlinearities at the vicinity of the crack front where the perfect electric displacement and magnetic induction saturations are assumed. The final formulation is discretized as a system of linear equations and an iterative approach is introduced to solve the unknown sizes of the two saturation zones by requiring that the electric displacement and magnetic induction intensity factors vanish at ends of their corresponding zones. The effect of the electric and magnetic fields on the saturation zones and the influence of the saturation zones on the stress intensity factor are illustrated with numerical examples.

Iterative Splitting Methods for Integrodifferential Equations: Theory and Applications

We present novel iterative splitting methods to solve integrodifferential equations. Such integrodifferential equations are applied, for example, in scattering problems of plasma simulations. We concentrate on a linearised integral part and a reformulation to a system of first order differential equations. Such modifications allow for applying standard iterative splitting schemes and for extending the schemes, respecting the integral operator. A numerical analysis is presented of the system of semidiscretised differential equations as abstract Cauchy problems. In the applications, we present benchmark and initial realistic applications to transport problems with scattering terms. We also discuss the benefits of such iterative schemes as fast solver methods.

THE ITERATED PROJECTION METHOD FOR INTEGRO-DIFFERENTIAL EQUATIONS WITH CAUCHY KERNEL

In this paper we propose the iterated projection method for the approximate solution of an integro-differential equations with Cauchy kernel in <TEX>$L^2([-1,1</TEX><TEX>]</TEX><TEX>,\mathbb{C})$</TEX> using Legendre polynomials. We prove the convergence of the method. A system of linear equations is to be solved. Numerical examples illustrate the theoretical results.

Compact finite difference methods for high order integro-differential equations

High order integro-differential equations (IDEs), especially nonlinear, are usually difficult to solve even numerically. In this chapter, we give a highly accurate compact finite difference method to efficiently solve integro-differential equations, including high order and nonlinear problems. By numerical experiments, we show that compact finite difference method for integro-differential equations is fast and can obtain a high accuracy.