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1
Content available Finite difference method for the fractional order pseudo telegraph integro-differential equation
Modanli Mahmut, Ozbag Fatih, Akgül Ali
Journal of Applied Mathematics and Computational Mechanics
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2022
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Vol. 21, nr 1
41--54
EN
The main goal of this paper is to investigate the numerical solution of the fractional order pseudo telegraph integro-differential equation. We establish the first order finite difference scheme. Then for the stability analysis of the constructed difference scheme, we give theoretical statements and prove them. We also support our theoretical statements by performing numerical experiments for some fractions of order α.
2
Content available Dynamic vibration extinguished on a viscously elastic base
Ishmamatov M., Kulmuratov N., Khalilov S., Akhmedov N.
International Journal of Applied Mechanics and Engineering
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2021
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Vol. 26, no. 2
1--10
EN
The aim of the work is to develop algorithms and a set of programs for studying the dynamic characteristics of viscoelastic thin plates on a deformable base on which it is installed with several dynamic dampers. The theory of thin plates is used to obtain the equation of motion for the plate. The relationship between the efforts and the stirred plate obeys in the hereditary Boltzmann Voltaire integral. With this, a system of integro-differential equations is obtained which is solved by the method of complex amplitudes. As a result, a transcendental algebraic equation was obtained to determine the resonance frequencies, which is solved numerically by the Muller method. To determine the displacement of the point of the plate with periodic oscillations of the base of the plate, a linear inhomogeneous algebraic equation was obtained, which is solved by the Gauss method. The amplitude - frequency response of the midpoint of the plate is constructed with and without regard to the viscosity of the deformed element. The dependence of the stiffness of a deformed element on the frequency of external action is obtained to ensure optimal damping of vibrational vibrations of the plate.
3
Content available Integro-differential form of the first-order dual phase lag heat transfer equation and its numerical solution using the Control Volume Method
Ciesielski M., Mochnacki B., Majchrzak E.
Archives of Mechanics
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2020
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Vol. 72, nr 5
415--444
EN
The start point of the dual phase lag equation (DPLE) formulation is the generalized Fourier law in which two positive constants (the relaxation and thermalization times) appear. This type of equation can be used (among others) to describe the heat conduction processes proceeding in micro-scale. Depending on the number of components in the development of the generalized Fourier law into a power series, one can obtain both the first-order DPLE and the second-order one. In this paper the first-order dual phase lag equation is considered. The primary objective of this research is the transformation of DPLE differential form to the integro-differential one supplemented by the appropriate boundary-initial conditions. The obtained form of the differential equation is much simpler and more convenient at the stage of numerical computations – the numerical algorithm based on the three-time-level scheme reduces to the two-time-level one. To find the numerical solution, the Control Volume Method is used (the heating of thin metal film subjected to a laser beam is considered). The choice of the numerical method was not accidental. The method has a simple physical interpretation ensuring the preservation of the local and global energy balances. To our knowledge, it has not been used so far in this type of tasks. In the final part of the paper the examples of numerical simulations are presented and the conclusions are formulated.
4
Content available Minimizing the time spent in an interval by a Wiener process with uniform jumps
Lefebvre Mario
Control and Cybernetics
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2019
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Vol. 48, No. 3
407--415
EN
Let Xu(t) be a controlled Wiener process with jumps that are uniformly distributed over the interval [−c, c]. The aim is to minimize the time spent by Xu(t) in the interval [a, b]. The integro- differential equation, satisfied by the value function, is transformed into an ordinary differential equation and is solved explicitly for a particular case. The approximate solution obtained is precise when c is small.
5
Content available Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems
Denton Z., Ramirez J. D.
Opuscula Mathematica
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2017
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Vol. 37, no. 5
705--724
EN
In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example.
6
Content available Numerical computations of the fractional derivative in IVPS, examples in MATLAB and Mathematica
Sowa M.
Informatyka, Automatyka, Pomiary w Gospodarce i Ochronie Środowiska
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2017
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T. 7, nr 3
19--22
EN
The paper concerns a numerical method that deals with the computations of the fractional derivative in Caputo and Riemann-Liouville definitions. The method can be applied in time stepping processes of initial value problems. The name of the method is SubIval, which is an acronym of its previous name – the subinterval-based method. Its application in solving systems of fractional order state equations is presented. The method has been implemented into an ActiveX DLL. Exemplary MATLAB and Mathematica codes are given, which provide guidance on how the DLL can be used.
PL
Artykuł dotyczy numerycznej metody, którą wykorzystać można do obliczeń pochodnej ułamkowego rzędu w definicji Caputo i Riemanna-Liouville’a. Metoda ta może być wykorzystana przy rozwiązywaniu zagadnień początkowych. Metoda nosi nazwę SubIval, co jest akronimem jej poprzedniej, anglojęzycznej nazwy „subinterval-based method” (metoda podprzedziałów). Przedstawiono jej zastosowanie w rozwiązywaniu równań stanu ułamkowego rzędu. Metoda została zaimplementowana w bibliotece DLL z obsługą ActiveX. Przedstawiono przykładowe kody obliczeniowe (w oprogramowaniach MATLAB i Mathematica), które zawierają wskazówki dotyczące zastosowania biblioteki.
7
Content available On the summability of divergent power series solutions for certain first-order linear PDES
Hibino M.
Opuscula Mathematica
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2015
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Vol. 35, no. 5
595--624
EN
This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations.
8
Content available remote On existence of solutions of impulsive nonlinear functional neutral integro-differential equations with nonlocal condition
Jain R. S., Dhakne M. B.
Demonstratio Mathematica
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2015
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Vol. 48, nr 3
413--423
EN
In the present paper, we investigate the existence, uniqueness and continuous dependence of mild solutions of an impulsive neutral integro-differential equations with nonlocal condition in Banach spaces. We use Banach contraction principle and the theory of fractional power of operators to obtain our results.
9
Content available remote On mild solutions of nonlocal second order semilinear functional integro-differential equations
Jain R. S., Dhakne M. B.
Fasciculi Mathematici
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2014
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Nr 53
65--75
EN
In the present paper, we investigate the existence, uniqueness and continuous dependence on initial data of mild solutions of second order nonlocal semilinear functional integro-differential equations of more general type with delay in Banach spaces. Our analysis is based on the theory of strongly continuous cosine family of operators and modified version of Banach contraction theorem.
10
Content available On optimal boundary and distributed control of partial integro-differential equations
Khurshudyan A. Zh.
Archives of Control Sciences
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2014
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Vol. 24, no. 1
5-25
EN
A method of optimal control problems investigation for linear partial integro-differential equations of convolution type is proposed, when control process is carried out by boundary functions and right hand side of equation. Using Fourier real generalized integral transform control problem solution is reduced to minimization procedure of chosen optimality criterion under constraints of equality type on desired control function. Optimality of control impacts is obtained for two criteria, evaluating their linear momentum and total energy. Necessary and sufficient conditions of control problem solvability are obtained for both criteria. Numerical calculations are done and control functions are plotted for both cases of control process realization.
11
Content available remote An existence theorem for solutions of an integro-differential equation in Banach spaces
Dutkiewicz A.
Demonstratio Mathematica
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2012
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Vol. 45, nr 4
895-899
EN
The paper contains an existence theorem for local solutions of an initial value problem for a nonlinear integro-differential equation in Banach spaces. The assumptions and proofs are expressed in terms of measures of noncompactness.
12
Content available remote Application of muscle model to the musculoskeletal modeling
Wojnicz W., Wittbrodt E.
Acta of Bioengineering and Biomechanics
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2012
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Vol. 14, no. 3
29-39
EN
The purpose of this paper is to investigate new fusiform muscle models. Each of these models treats a muscle as a system composed of parts characterized by different mechanical properties. These models explain the influence of differences in the stiffness of lateral parts and the degree of muscle model discretization. Each muscle model is described by a system of differential equations and a single integro-differential equation. Responses of fifty-four muscle model forms are examined using a complex exertion composed of three types: eccentric-concentric exertion, isokinetic-isometric exertion and step exertion.
13
Content available remote Initial value problems for second-order integro-differential equations on unbounded domains in Banach space
Liu Y.
Demonstratio Mathematica
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2005
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Vol. 38, nr 2
349--364
EN
In this paper, the Monch fixed point theorem is used to investigate the existence of solutions of initial value problem (IVP, for short) for second order nonlinear integro-differential equations on infinite intervals in a Banach space. At the same time, the uniqueness of solution for IVP is obtained also.
14
Content available remote A note on the occurrence of integro-differential equations in engineering mechanics
Pala Y., Karadere G.
International Journal of Applied Mechanics and Engineering
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2003
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Vol. 8, no 3
503-506
EN
The study of dynamics of packages of mass m delivered from a conveyor to a smooth circular ramp of radius r is important in transport technology and engineering dynamics. The present paper is devoted to reanalysing this problem since the inclusion of frictional forces results in an integro-differential equation that in general needs the utilisation of Neumann-series. The integro-differential equation thus obtained is transformed into a first order differential equation that can easily be solved analytically.
15
Content available remote On an initial value problem for singular integro-differential equations
Šmarda Z.
Demonstratio Mathematica
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2002
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Vol. 35, nr 4
803-811
EN
The purpose of this paper is to study the existence and asymptotic behaviour of solutions of a nonlinear singular integro-differential equation.
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