Abstract: This paper compared the efficiency of Simpson’s 1/3 and 3/8 rules for the numerical solution of first order Volterra integro-differential equations. In developing the solution, collocation approximation method was adopted using the shifted Legendre polynomial as basis function. A block method approach is preferred to the predictor corrector method for being self-starting. Experimental results confirmed that the Simpson’s 3/8 rule is more efficient than the Simpson’s 1/3 rule.
Abstract: Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations.
Abstract: In this paper, the influence of van der Waals, as well as electrostatic forces on the structural behavior of MEMS and NEMS actuators, has been investigated using of a Euler-Bernoulli beam continuous model. In the proposed nonlinear model, the electrostatic fringing-fields and the mid-plane stretching (geometric nonlinearity) effects have been considered. The nonlinear integro-differential equation governing the static structural behavior of the actuator has been derived. An original Galerkin-based reduced-order model has been developed to avoid problems arising from the nonlinearities in the differential equation. The obtained reduced-order model equations have been solved numerically using the Newton-Raphson method. The basic design parameters such as the pull-in parameters (voltage and deflection at pull-in), as well as the detachment length due to the van der Waals force of some investigated micro- and nano-actuators have been calculated. The obtained numerical results have been compared with some other existing methods (finite-elements method and finite-difference method) and the comparison showed good agreement among all assumed numerical techniques.
Abstract: In this paper, the numerical solution of optimal control problem (OCP) for systems governed by Volterra integro-differential
(VID) equation is considered. The method is developed by means
of the Legendre wavelet approximation and collocation method. The
properties of Legendre wavelet together with Gaussian integration
method are utilized to reduce the problem to the solution of nonlinear
programming one. Some numerical examples are given to confirm the
accuracy and ease of implementation of the method.
Abstract: In this paper, Semi-orthogonal B-spline scaling
functions and wavelets and their dual functions are presented
to approximate the solutions of integro-differential equations.The
B-spline scaling functions and wavelets, their properties and the
operational matrices of derivative for this function are presented to
reduce the solution of integro-differential equations to the solution of
algebraic equations. Here we compute B-spline scaling functions of
degree 4 and their dual, then we will show that by using them we have
better approximation results for the solution of integro-differential
equations in comparison with less degrees of scaling functions
Abstract: In this paper, by establishing a new comparison result, we investigate the existence of positive solutions for initial value problems of nonlinear systems of second order integro-differential equations in Banach space.We improve and generalize some results (see[5,6]), and the results is new even in finite dimensional spaces.
Abstract: The objective of this paper is to analyse the
application of the Half-Sweep Gauss-Seidel (HSGS) method by using
the Half-sweep approximation equation based on central difference
(CD) and repeated trapezoidal (RT) formulas to solve linear fredholm
integro-differential equations of first order. The formulation and
implementation of the Full-Sweep Gauss-Seidel (FSGS) and Half-
Sweep Gauss-Seidel (HSGS) methods are also presented. The HSGS
method has been shown to rapid compared to the FSGS methods.
Some numerical tests were illustrated to show that the HSGS method
is superior to the FSGS method.
Abstract: In this article, an adaptive least-squares mixed finite element method is studied for pseudo-parabolic integro-differential equations. The solutions of least-squares mixed weak formulation and mixed finite element are proved. A posteriori error estimator is constructed based on the least-squares functional and the posteriori errors are obtained.
Abstract: Beginning from the creator of integro-differential
equations Volterra, many scientists have investigated these
equations. Classic method for solving integro-differential
equations is the quadratures method that is successfully applied up
today. Unlike these methods, Makroglou applied hybrid methods
that are modified and generalized in this paper and applied to the
numerical solution of Volterra integro-differential equations. The
way for defining the coefficients of the suggested method is also
given.
Abstract: Solution of some practical problems is reduced to the
solution of the integro-differential equations. But for the numerical
solution of such equations basically quadrature methods or its
combination with multistep or one-step methods are used. The
quadrature methods basically is applied to calculation of the integral
participating in right hand side of integro-differential equations. As
this integral is of Volterra type, it is obvious that at replacement with
its integrated sum the upper limit of the sum depends on a current
point in which values of the integral are defined. Thus we receive the
integrated sum with variable boundary, to work with is hardly.
Therefore multistep method with the constant coefficients, which is
free from noted lack and gives the way for finding it-s coefficients is
present.
Abstract: In this paper, a nonconforming mixed finite element method is studied for semilinear pseudo-hyperbolic partial integrodifferential equations. By use of the interpolation technique instead of the generalized elliptic projection, the optimal error estimates of the corresponding unknown function are given.
Abstract: In this paper the Laplace Decomposition method is developed to solve linear and nonlinear fractional integro- differential equations of Volterra type.The fractional derivative is described in the Caputo sense.The Laplace decomposition method is found to be fast and accurate.Illustrative examples are included to demonstrate the validity and applicability of presented technique and comparasion is made with exacting results.
Abstract: In this paper, some new nonlinear generalized
Gronwall-Bellman-Type integral inequalities with mixed time delays
are established. These inequalities can be used as handy tools
to research stability problems of delayed differential and integral
dynamic systems. As applications, based on these new established
inequalities, some p-stable results of a integro-differential equation
are also given. Two numerical examples are presented to illustrate
the validity of the main results.
Abstract: In this article two algorithms, one based on variation iteration method and the other on Adomian's decomposition method, are developed to find the numerical solution of an initial value problem involving the non linear integro differantial equation where R is a nonlinear operator that contains partial derivatives with respect to x. Special cases of the integro-differential equation are solved using the algorithms. The numerical solutions are compared with analytical solutions. The results show that these two methods are efficient and accurate with only two or three iterations
Abstract: This paper evaluates the dividend payments for general
claim size distributions in the presence of a dividend barrier. The
surplus of a company is modeled using the classical risk process
perturbed by diffusion, and in addition, it is assumed to accrue interest
at a constant rate. After presenting the integro-differential equation
with initial conditions that dividend payments satisfies, the paper
derives a useful expression of the dividend payments by employing
the theory of Volterra equation. Furthermore, the optimal value of
dividend barrier is found. Finally, numerical examples illustrate the
optimality of optimal dividend barrier and the effects of parameters
on dividend payments.