Solution Of Integro-differential Equation Research Articles

Bending-torsional vibrations of aircraft wing

The problems of bending-torsional flutter of an aircraft wing are considered. Using equations of motion by the Bubnov-Galerkin method, based on the polynomial approximation of deflections and angle of twist, the problem is reduced to studying a system of ordinary integro-differential equations, where the independent variable is time. The solution of integro-differential equations with singular kernels is found by a numerical method based on quadrature formulas. The influence of physical-mechanical and geometric parameters on the flutter of the aircraft wing was studied. It was established that an account for the viscoelastic properties of the material of thin-walled aircraft structures leads to a 40-60% decrease in the critical flutter velocity. It is shown that an increase in the elongation parameter of an aircraft wing leads to a decrease in the flutter velocity.

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.

Hyers-Ulam stability of fractional integro-differential equation with a positive constant coefficient involving the generalized Caputo fractional derivative

The purpose of this paper is to investigate the existence and uniqueness of a solution, and the continuous dependence on the input data of the solution of integro-differential equations with a positive constant coefficient involving fractional order derivative (FIDEs). In addition, we also provide the sufficient conditions for the Hyers-Ulam stability (HU-stability) and the Hyers-Ulam-Rassias stability (HUR-stability) of FIDEs. Finally, the HUR-stability of the well-known model of RLC circuit in the form of FIDEs is also surveyed.

Mathematical modeling parametric vibrations of the pipeline with pulsating fluid flow

A mathematical model of the dynamics of a straight viscoelastic pipe conveying pulsating fluid has been developed in the paper. Using the Bubnov-Galerkin method, a mathematical model of the problem is reduced to solving a system of ordinary integro-differential equations, where time is an independent variable. Solution of integro-differential equations is determined by a numerical method based on the elimination of a singularity in the relaxation kernel of an integral operator. A system of algebraic equations is obtained according to the numerical method for unknowns. To solve the system of algebraic equations, the Gauss method is used. A computational algorithm is developed for solving the problems of the dynamics of viscoelastic pipelines conveying fluid.

Inhomogeneous parametric scaling and variable-order fractional diffusion equations

We discuss the derivation and the solutions of integrodifferential equations (variable-order time-fractional diffusion equations) following as continuous limits for lattice continuous time random walk schemes with power-law waiting-time probability density functions whose parameters are position-dependent. We concentrate on subdiffusive cases and discuss two situations as examples: A system consisting of two parts with different exponents of subdiffusion, and a system in which the subdiffusion exponent changes linearly from one end of the interval to another one. In both cases we compare the numerical solutions of generalized master equations describing the process on the lattice to the corresponding solutions of the continuous equations, which follow by exact solution of the corresponding equations in the Laplace domain with subsequent numerical inversion using the Gaver-Stehfest algorithm.

Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

In this paper, we introduce a new scheme based on the exponential spline function for solving linear second-order Fredholm integro-differential equations. Our approach consists of reducing the problem to a set of linear equations. We prove the convergence analysis of the method applied to the solution of integro-differential equations. The method is described and illustrated with numerical examples. The results reveal that the method is accurate and easy to apply. Moreover, results are compared with the method in (J. Comput. Appl. Math. 290:633–640, 2015).

О построении функциональных полиномов для решений интегродифференциальных уравнений

О построении функциональных полиномов для решений интегродифференциальных уравнений

Periodic Solution of Integro-Differential Equations Depended on Special Functions with Singular Kernels and Boundary Integral Conditions

Periodic Solution of Integro-Differential Equations Depended on Special Functions with Singular Kernels and Boundary Integral Conditions

Convergence of numerical schemes for the solution of partial integro-differential equations used in heat transfer

Integro-differential equations play an important role in may physical phenomena. For instance, it appears in fields like fluid dynamics, biological models and chemical kinetics. One of the most important physical applications is the heat transfer in heterogeneous materials, where physician are looking for efficient methods to solve their modeled equations. The difficulty of solving integro-differential equations analytically made mathematician to search about efficient methods to find an approximate solution. The present article is designed to supply numerical solution of a parabolic Volterra integro-differential equation under initial and boundary conditions. We have made an attempt to develop a numerical solution via the use of Sinc-Galerkin method, the convergence analysis via the use of fixed point theory has been discussed, and showed to be of exponential order. For comparison purposes, we approximate the solution of integro-differential equation using Adomian decomposition method. Sometimes, the Adomian decomposition method is a highly efficient technique used to approximate analytical solution of differential equations, applicability of Adomian decomposition method to partial integro-differential equations has not been studied in details previously in the literatures. In addition, we present numerical examples and comparisons to support the validity of these proposed methods.

Computer simulation of oscillatory processes of viscoelastic elements of thin-walled structures in a gas flow

Results of numerical investigation of dynamic behavior of deformed wing aircraft in a gas flow are presented in the paper. Vibrations with respect to deflections are described by a system of integro-differential equations in partial derivatives. Using the Bubnov-Galerkin method, the problem is reduced to a system of ordinary integro-differential equations, where time is an independent variable. The solutions of integro-differential equations are determined by a numerical method based on the use of quadrature formulas. Computational algorithms and a package of applied programs have been created to solve problems on nonlinear flutter of viscoelastic elements of an aircraft. The reliability of the solution of the problem is confirmed by comparison with known numerical and analytical results. The effect of different boundary conditions on critical flutter velocity is studied. Critical velocity and critical flutter time of viscoelastic plates are determined. It is shown that the singularity parameter α affects not only the vibrations of viscoelastic systems, but also critical time and critical flutter velocity. It is stated that consideration of viscoelastic properties of plate material leads to 40 60% decrease in critical flutter velocity.

Nonlinear supersonic flutter for the viscoelastic orthotropic cylindrical shells in supersonic flow

On the example of an orthotropic shell, the problems of the dynamics of thin-walled structures under aerodynamic loading are studied, taking into account the viscoelastic properties of material and geometric nonlinearity. The aerodynamic pressure is determined using the AA. Ilyushin's piston theory. Equations of motion relative to displacements are described by a system of integro-differential equations in partial derivatives. Using the Bubnov–Galerkin method, based on the polynomial approximation of deflections, the problem is reduced to a system of ordinary integro-differential equations, where time is an independent variable. Solutions of integro-differential equations are determined by a numerical method based on the elimination of the singularity in the relaxation kernel of the integral operator. Computational algorithms and applied programs have been developed to solve the problems on the nonlinear flutter for viscoelastic elements of an aircraft. The critical flutter speed for the viscoelastic orthotropic cylindrical shells is determined. It is established that an account of viscoelastic properties of shell material leads to a decrease in the critical flutter.

Mathematical simulation of nonlinear oscillations of viscoelastic pipelines conveying fluid

• A mathematical model of the problem of nonlinear oscillations of a viscoelastic pipeline conveying fluid is developed. • A computational algorithm is developed to solve the problems of the dynamics of viscoelastic pipelines. • Numerically investigated the influence of singularity in the heredity kernels on vibrations of the pipeline. A mathematical model of the problem of nonlinear oscillations of a viscoelastic pipeline conveying fluid is developed in the paper. The Boltzmann–Volterra integral model with weakly singular kernels of heredity is used to describe the processes of pipeline strain. Using the Bubnov–Galerkin method, the mathematical model of the problem is reduced to the study of a system of ordinary integro-differential equations, where time is an independent variable. The solution of integro-differential equations is determined by a numerical method based on the elimination of the singularity in the relaxation kernel of the integral operator. Using the numerical method for unknowns, a system of algebraic equations is obtained. To solve a system of algebraic equations, the Gauss method is used. A computational algorithm is developed to solve the problems of the dynamics of viscoelastic pipelines with a flowing fluid. The algorithm of the proposed method makes it possible to investigate in detail the effect of rheological parameters on the character of vibrational strength of viscoelastic pipelines with a fluid flow, in particular, in the study of free oscillations of pipelines based on the theory of ideally elastic shells. On the basis of the computational algorithm developed, a set of applied computer programs has been created, which makes it possible to carry out numerical studies of pipeline oscillations. The influence of singularity in the heredity kernels and the geometric parameters of the pipeline on the vibrations of structures possessing viscoelastic properties is numerically investigated. It is shown that an account of viscoelastic properties of pipeline material leads decrease in the amplitude and frequency of oscillation. It is established that to reveal the influence of viscoelastic properties of structure material on the pipeline oscillations, it is necessary to use the Abel-type weakly singular kernels of heredity. The obtained results of numerical simulation can be used in the enterprises of oil and gas industries, as well as in design organizations.

The analytical solutions for conformable integral equations and integro-differential equations by conformable Laplace transform

In this article, existence theorem for conformable Laplace transform is expressed. Then by using basic properties of conformable Laplace transform such as convolution theorem, conformable Laplace transform of fractional derivative and fractional integral, authors obtained the exact solution of initial value problems for integral equations and integro-differential equations where the derivatives and integrals are in conformable sense. In the literature it is the first time that obtaining the solutions of integro differential equations, integral equations by means of conformable fractional derivative.

Об одном методе численного решения интегро-дифференциальных уравнений

Об одном методе численного решения интегро-дифференциальных уравнений

Ruin probabilities in classical risk models with gamma claims

In this paper, we provide three equivalent expressions for ruin probabilities in a Cramér–Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag–Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.

Existence of solutions of abstract non-autonomous second order integro-differential equations

In this paper the existence of solutions of a non-autonomous abstract integro-differential equation of second order is considered. Assuming the existence of an evolution operator corresponding to the associate abstract non-autonomous Cauchy problem of second order, we establish the existence of a resolvent operator for the homogeneous integro-differential equation and the existence of mild solutions to the inhomogeneous integro-differential equation. Furthermore, we study the existence of classical solutions of the integro-differential equation. Finally, we apply our results to the study of the existence of solutions of a non-autonomous wave equation.

A numerical method for solution of integro-differential equations of fractional order

Bu calismada, sinc siralama yontemi kesirli mertebeden Volterra integro-diferansiyel denklemleri yaklasik olarak cozmek icin gelistirilmistir. Kesirli turev, kesirli analizde sikca kullanilan Caputo anlaminda tanimlanmistir. Elde edilen sonuclar iki yeni teorem ile verilmistir. Bazi sayisal ornekleri teorik sonuclari gostermek icin sunulmustur.

Invariant and partially invariant solutions of integro-differential equations for linear thermoviscoelastic aging materials with memory

A linear thermoviscoelastic model for homogeneous, aging materials with memory is established. A system of integro-differential equations is obtained by using two motions (a one-dimensional motion and a shearing motion) for this model. Applying the group analysis method to the system of integro-differential equations, the admitted Lie group is determined. Using this admitted Lie group, invariant and partially invariant solutions are found. The present paper gives a first example of application of partially invariant solutions to integro-differential equations.

Viscosity Solutions of Integro-Differential Equations for Nonruin Probabilities

We consider a model of an insurance company investing its reserve into a risky asset whose price follows a geometric Levy process. We show that the nonruin probability is a viscosity solution of a second order integro-differential equation and prove a uniqueness theorem for the latter.

Existence and Exponential Stability of Weighted Pseudo Almost Periodic Classical Solutions of Integro-Differential Equations with Analytic Semigroups

In this paper, we investigate a class of integro-differential equations having analytic semigroups given by $$\begin{aligned} x'(t)=Ax(t)+f(t,x(t),Gx(t)). \end{aligned}$$ Our main results concern the existence, uniqueness and globally exponential stability of weighted pseudo-almost periodic classical solutions. An example is given to illustrate our results.