Search results for: Volterra integro-differential equation.

Authors: Fuziyah Ishak, Siti Norazura Ahmad

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.

Keywords: Accuracy, extended trapezoidal method, numerical solution, Volterra integro-differential equations.

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Authors: G.Mehdiyeva, V.Ibrahimov, M.Imanova

Abstract:

As is known, one of the priority directions of research works of natural sciences is introduction of applied section of contemporary mathematics as approximate and numerical methods to solving integral equation into practice. We fare with the solving of integral equation while studying many phenomena of nature to whose numerically solving by the methods of quadrature are mainly applied. Taking into account some deficiency of methods of quadrature for finding the solution of integral equation some sciences suggested of the multistep methods with constant coefficients. Unlike these papers, here we consider application of hybrid methods to the numerical solution of Volterra integral equation. The efficiency of the suggested method is proved and a concrete method with accuracy order p = 4 is constructed. This method in more precise than the corresponding known methods.

Keywords: Volterra integral equation, hybrid methods, stability and degree, methods of quadrature

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Authors: Khosrow Maleknejad, Asyieh Ebrahimzadeh

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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.

Keywords: Collocation method, Legendre wavelet, optimal control, Volterra integro-differential equation.

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Authors: Li-Li Li, Jinghai Feng, Lixin Song

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.

Keywords: Dividend payout, Integro-differential equation, Jumpdiffusion model, Volterra equation

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Authors: M.Imanova, G.Mehdiyeva, V.Ibrahimov

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.

Keywords: Volterra integro-differential equations, multistepmethods, finite-difference methods, initial value problem

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Authors: Rabha W. Ibrahim

Abstract:

Our main aim in this paper is to use the technique of non expansive operators to more general iterative and non iterative fractional differential equations (Cauchy type ). The non integer case is taken in sense of Riemann-Liouville fractional operators. Applications are illustrated.

Keywords: Fractional calculus, fractional differential equation, Cauchy equation, Riemann-Liouville fractional operators, Volterra integral equation, non-expansive mapping, iterative differential equation.

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Authors: M. B. Meenavathi, K. Rajesh

Abstract:

In this paper, we propose a new class of Volterra series based filters for image enhancement and restoration. Generally the linear filters reduce the noise and cause blurring at the edges. Some nonlinear filters based on median operator or rank operator deal with only impulse noise and fail to cancel the most common Gaussian distributed noise. A class of second order Volterra filters is proposed to optimize the trade-off between noise removal and edge preservation. In this paper, we consider both the Gaussian and mixed Gaussian-impulse noise to test the robustness of the filter. Image enhancement and restoration results using the proposed Volterra filter are found to be superior to those obtained with standard linear and nonlinear filters.

Keywords: Gaussian noise, Image enhancement, Imagerestoration, Linear filters, Nonlinear filters, Volterra series.

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Authors: M. B. Meenavathi, K. Rajesh

Abstract:

Color image segmentation plays an important role in computer vision and image processing areas. In this paper, the features of Volterra filter are utilized for color image segmentation. The discrete Volterra filter exhibits both linear and nonlinear characteristics. The linear part smoothes the image features in uniform gray zones and is used for getting a gross representation of objects of interest. The nonlinear term compensates for the blurring due to the linear term and preserves the edges which are mainly used to distinguish the various objects. The truncated quadratic Volterra filters are mainly used for edge preserving along with Gaussian noise cancellation. In our approach, the segmentation is based on K-means clustering algorithm in HSI space. Both the hue and the intensity components are fully utilized. For hue clustering, the special cyclic property of the hue component is taken into consideration. The experimental results show that the proposed technique segments the color image while preserving significant features and removing noise effects.

Keywords: Color image segmentation, HSI space, K–means clustering, Volterra filter.

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Authors: Rabha W. Ibrahim

Abstract:

We study a Dirichlet boundary value problem for Lane-Emden equation involving two fractional orders. Lane-Emden equation has been widely used to describe a variety of phenomena in physics and astrophysics, including aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres,and thermionic currents. However, ordinary Lane-Emden equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractalmedium, numerous generalizations of Lane-Emden equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Lane-Emden equation. This gives rise to the fractional Lane-Emden equation with a single index. Recently, a new type of Lane-Emden equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskiis fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space. Ulam-Hyers stability for iterative Cauchy fractional differential equation is defined and studied.

Keywords: Fractional calculus, fractional differential equation, Lane-Emden equation, Riemann-Liouville fractional operators, Volterra integral equation.

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Authors: G.Mehdiyeva, M.Imanova, V.Ibrahimov

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.

Keywords: Integro-differential equations, initial value problem, hybrid methods, predictor-corrector method

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Authors: Yongkun Li, Kaihong Zhao

Abstract:

By using Mawhin-s continuation theorem of coincidence degree theory, we establish the existence of 2n positive periodic solutions for n species non-autonomous Lotka-Volterra competition systems with harvesting terms. An example is given to illustrate the effectiveness of our results.

Keywords: Positive periodic solutions, Lotka-Volterra competition system, coincidence degree, harvesting term.

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Authors: J. B. Seo, K. J. Kim, S. W. Nam

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In this paper, a nonlinear acoustic echo cancellation (AEC) system is proposed, whereby 3rd order Volterra filtering is utilized along with a variable step-size Gauss-Seidel pseudo affine projection (VSSGS-PAP) algorithm. In particular, the proposed nonlinear AEC system is developed by considering a double-talk situation with near-end signal variation. Simulation results demonstrate that the proposed approach yields better nonlinear AEC performance than conventional approaches.

Keywords: Acoustic echo cancellation (AEC), Volterra filtering, variable step-size, GS-PAP.

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Authors: Zixin Liu, Shu Lü, Shouming Zhong, Mao Ye

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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.

Keywords: Gronwall-Bellman-Type integral inequalities, integrodifferential equation, p-exponentially stable, mixed delays.

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Authors: Yongkun Li, Erliang Xu

Abstract:

In this paper, by using Mawhin-s continuation theorem of coincidence degree theory, we establish the existence of multiple positive periodic solutions of a competitor-competitor-mutualist Lotka-Volterra system with harvesting terms. Finally, an example is given to illustrate our results.

Keywords: Positive periodic solutions, competitor-competitor mutualist Lotka-Volterra systems, coincidence degree, harvesting term.

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Authors: Kaihong Zhao

Abstract:

In this paper, the existence of 2n positive periodic solutions for n species non-autonomous Lotka-Volterra cooperative systems with harvesting terms is established by using Mawhin-s continuation theorem of coincidence degree theory and matrix inequality. An example is given to illustrate the effectiveness of our results.

Keywords: Multiple positive periodic solutions, Nonautonomous Lotka-Volterra cooperative system, Coincidence degree, Harvesting term.

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Authors: Aziz Sezgin

Abstract:

We consider the problem of stabilization of an unstable heat equation in a 2-D, 3-D and generally n-D domain by deriving a generalized backstepping boundary control design methodology. To stabilize the systems, we design boundary backstepping controllers inspired by the 1-D unstable heat equation stabilization procedure. We assume that one side of the boundary is hinged and the other side is controlled for each direction of the domain. Thus, controllers act on two boundaries for 2-D domain, three boundaries for 3-D domain and ”n” boundaries for n-D domain. The main idea of the design is to derive ”n” controllers for each of the dimensions by using ”n” kernel functions. Thus, we obtain ”n” controllers for the ”n” dimensional case. We use a transformation to change the system into an exponentially stable ”n” dimensional heat equation. The transformation used in this paper is a generalized Volterra/Fredholm type with ”n” kernel functions for n-D domain instead of the one kernel function of 1-D design.

Keywords: Backstepping, boundary control, 2-D, 3-D, n-D heat equation, distributed parameter systems.

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Authors: Yongzhi Liao

Abstract:

By using the theory of exponential dichotomy and Banach fixed point theorem, this paper is concerned with the problem of the existence and uniqueness of positive almost periodic solution in a delayed Lotka-Volterra recurrent neural networks with harvesting terms. To a certain extent, our work in this paper corrects some result in recent years. Finally, an example is given to illustrate the feasibility and effectiveness of the main result.

Keywords: positive almost periodic solution, Lotka-Volterra, neural networks, Banach fixed point theorem, harvesting

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Authors: N. M. Kamoh, D. G. Gyemang, M. C. Soomiyol

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.

Keywords: Collocation shifted Legendre polynomials, Simpson’s rule and Volterra integro-differential equations.

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Authors: N. Ebrahimi, J. Rashidinia

Abstract:

In this paper, numerical solution of system of Fredholm and Volterra integral equations by means of the Spline collocation method is considered. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The solution is collocated by cubic B-spline and the integrand is approximated by the Newton-Cotes formula. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods to illustrate the accuracy and the implementation of our method.

Keywords: Convergence analysis, Cubic B-spline, Newton- Cotes formula, System of Fredholm and Volterra integral equations.

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Authors: Hirotaka Watanabe, Takaaki Kondo, Daiki Yoshida, Ariyoshi Nakayama, Taichi Sato, Shuhei Kuriyama, Hiroyuki Kamata

Abstract:

In this paper, we propose a chaotic cipher system consisting of Improved Volterra Filters and the mapping that is created from the actual voice by using Radial Basis Function Network. In order to achieve a practical system, the system supposes to use the digital communication line, such as the Internet, to maintain the parameter matching between the transmitter and receiver sides. Therefore, in order to withstand the attack from outside, it is necessary that complicate the internal state and improve the sensitivity coefficient. In this paper, we validate the robustness of proposed method from three perspectives of "Chaotic properties", "Randomness", "Coefficient sensitivity".

Keywords: Chaos cipher, 16-bit-length fixed point arithmetic, Volterra filter, Seacret communications, RBF Network

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Authors: Changqing Yang, Jianhua Hou

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.

Keywords: Integro-differential equations, Laplace transform, fractional derivative, adomian polynomials, pade appoximants.

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Authors: Rudolf Csikja, Janos Toth

Abstract:

Methods to detect and localize time singularities of polynomial and quasi-polynomial ordinary differential equations are systematically presented and developed. They are applied to examples taken form different fields of applications and they are also compared to better known methods such as those based on the existence of linear first integrals or Lyapunov functions.

Keywords: blow up, finite escape time, polynomial ODE, singularity, Lotka–Volterra equation, Painleve analysis, Ψ-series, global existence

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Authors: Ahmet Tekcan, Betül Gezer, Osman Bizim

Abstract:

Let k ≥ 1 and t ≥ 0 be two integers and let d = k2 + k be a positive non-square integer. In this paper, we consider the integer solutions of Pell equation x2 - dy2 = 2t. Further we derive a recurrence relation on the solutions of this equation.

Keywords: Pell equation, Diophantine equation.

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Authors: A. Zerarka, A. Soukeur, N. Khelil

Abstract:

In the present work, we introduce the particle swarm optimization called (PSO in short) to avoid the Runge-s phenomenon occurring in many numerical problems. This new approach is tested with some numerical examples including the generalized integral quadrature method in order to solve the Volterra-s integral equations

Keywords: Integral equation, particle swarm optimization, Runge's phenomenon.

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Authors: Armend Sh. Shabani

Abstract:

Let D ≠ 1 be a positive non-square integer. In this paper are given the proofs for two conjectures related to Pell-s equation x2 -Dy2 = ± 4, proposed by A. Tekcan.

Keywords: Pell's equation, solutions of Pell's equation.

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Authors: Javad Abdalkhani

Abstract:

Convergence of power series solutions for a class of non-linear Abel type equations, including an equation that arises in nonlinear cooling of semi-infinite rods, is very slow inside their small radius of convergence. Beyond that the corresponding power series are wildly divergent. Implementation of nonlinear sequence transformation allow effortless evaluation of these power series on very large intervals..

Keywords: Nonlinear transformation, Abel Volterra Equations, Mathematica

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Authors: Bi-Huei Tsai

Abstract:

This work explores the inter-region investment behaviors of Integrated Circuit (IC) design industry from Taiwan to China using the amount of foreign direct investment (FDI). According to the mutual dependence among different IC design industrial locations, Lotka-Volterra model is utilized to explore the FDI interactions between South and East China. Effects of inter-regional collaborations on FDI flows into China are considered. The analysis results show that FDIs into South China for IC design industry significantly inspire the subsequent FDIs into East China, while FDIs into East China for Taiwan’s IC design industry significantly hinder the subsequent FDIs into South China. Because the supply chain along IC industry includes upstream IC design, midstream manufacturing, as well as downstream packing and testing enterprises, IC design industry has to cooperate with IC manufacturing, packaging and testing industries in the same area to form a strong IC industrial cluster. Taiwan’s IC design industry implement the largest FDI amount into East China and the second largest FDI amount into South China among the four regions: North, East, Mid-West and South China. If IC design houses undertake more FDIs in South China, those in East China are urged to incrementally implement more FDIs into East China to maintain the competitive advantages of the IC supply chain in East China. On the other hand, as the FDIs in East China rise, the FDIs in South China will successively decline since capitals have concentrated in East China. In addition, this investigation proves that the prediction of Lotka-Volterra model in FDI trends is accurate because the industrial interactions between the two regions are included. Finally, this work confirms that the FDI flows cannot reach a stable equilibrium point, so the FDI inflows into East and South China will expand in the future.

Keywords: Lotka-Volterra model, Foreign direct investment, Competitive, Equilibrium analysis.

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Authors: Y. Pala, M. O. Ertas

Abstract:

In this paper, the general Riccati equation is analytically solved by a new transformation. By the method developed, looking at the transformed equation, whether or not an explicit solution can be obtained is readily determined. Since the present method does not require a proper solution for the general solution, it is especially suitable for equations whose proper solutions cannot be seen at first glance. Since the transformed second order linear equation obtained by the present transformation has the simplest form that it can have, it is immediately seen whether or not the original equation can be solved analytically. The present method is exemplified by several examples.

Keywords: Riccati Equation, ordinary differential equation, nonlinear differential equation, analytical solution, proper solution.

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Authors: Ahmet Tekcan, Arzu Özkoç, Canan Kocapınar, Hatice Alkan

Abstract:

Let p be a prime number such that p ≡ 1(mod 4), say p = 1+4k for a positive integer k. Let P = 2k + 1 and Q = k2. In this paper, we consider the integer solutions of the Pell equation x2-Py2 = Q over Z and also over finite fields Fp. Also we deduce some relations on the integer solutions (xn, yn) of it.

Keywords: Pell equation, solutions of Pell equation.

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Authors: Ahmet Tekcan, Arzu Özkoç, Hatice Alkan

Abstract:

In this work, we consider the number of integer solutions of Diophantine equation D : y2 - 2yx - 3 = 0 over Z and also over finite fields Fp for primes p ≥ 5. Later we determine the number of rational points on curves Ep : y2 = Pp(x) = yp 1 + yp 2 over Fp, where y1 and y2 are the roots of D. Also we give a formula for the sum of x- and y-coordinates of all rational points (x, y) on Ep over Fp.

Keywords: Diophantine equation, Pell equation, quadratic form.

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