This paper presents the application of homotopy perturbation and variational iteration methods as numerical methods for Fredholm integrodifferential equation of fractional order with initial-boundary conditions. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented. � 2012 Asma Ali Elbeleze et al.

Fractional partial differential equations arise from many fields of physics and apply a very important role in various branches of science and engineering. Finding accurate and efficient methods for solving partial differential equations of fractional order has become an active research undertaking. In the present paper, the homotopy perturbation method proposed by J-H He has been used to obtain the solution of some fractional partial differential equations with variable coefficients. Exact and/or approximate analytical solutions of these equations are obtained.

In the present paper, we study the integro-differential equations which are combination of differential and Fredholm-Volterra equations that have the fractional order with constant coefficients by the homotopy perturbation and the variational iteration. The fractional derivatives are described in Caputo sense. Some illustrative examples are presented. � 2015 The Authors.

In this note, we propose a new class of orthogonal polynomials (named Bachok-Hasham polynomials of the first and second kind for order k, denote it as Z(i,n)k(x), i={ 1,2 }, which is extension of the Chebyshev polynomials of the first and second kind respectively. It is found that Bachok - Hasham polynomials of first and second kind Z(i,n)k(x) are orthogonal with respect to weights w(1,k)(x)=xk-11-x2k, w(2,k)(x)=xk-11-x2k on the interval [-1,1], where k is positive odd integers. Spectral properties Bachok - Hasham polynomials of the first and second kind Z(i,n)k(x),i={ 1,2 } are proved. These properties are used to solve a special class of singular integral equations. Finally, numerical examples and comparison results with other methods are provided to illustrate the effectiveness and accuracy of the proposed method. � 2018 Author(s).

We are concerned here with singular partial differential equations of fractional order (FSPDEs). The variational iteration method (VIM) is applied to obtain approximate solutions of this type of equations. Convergence analysis of the VIM is discussed. This analysis is used to estimate the maximum absolute truncated error of the series solution. A comparison between the results of VIM solutions and exact solution is given. The fractional derivatives are described in Caputo sense. � 2014 Asma Ali Elbeleze et al.

Malwares attack such as by the worm, virus, trojan horse and botnet have caused lots of troublesome for many organisations and users which lead to the cybercrime. Living in a cyber world, being infected by these malwares becoming more common. Nowadays the malwares attack especially by the trojan horse is becoming more sophisticated and intelligent, makes it is harder to be detected than before. Therefore, in this research paper, a new model called Efficient Trojan Detection Model (ETDMo) is built to detect trojan horse attacks more efficiently. In this model, the static, dynamic and automated analyses were conducted and the machine learning algorithms were applied to optimize the performance. Based on the experiment conducted, the Sequential Minimal Optimization (SMO) algorithm has outperformed other machine learning algorithms with 98.2 % of true positive rate and with 1.7 % of false positive rate. � Springer International Publishing Switzerland 2015.

Nowadays, worms are becoming more sophisticated, intelligent and hard to be detected and responded than before and it becomes as one of the main issues in cyber security. It caused loss millions of money and productivities in many organizations and users all over the world. Currently, there are many works related with worm detection techniques but not much research is focusing on worm response. Therefore, in this research paper, a new model to respond to the worms attack efficiently is built. This worm response model is called as eZSiber, inspired by apoptosis or also known as cell-programmed death. It is a concept borrowed from human immunology system (HIS), where it has been mapped into network security environment. Once the user�s computer detects any indication of the worm attacks, the apoptosis is triggered. In order to trigger the apoptosis, security metrics plays a very important role in identifying the weight and the severity of the worm attacks. In this model, the static and dynamic analyses were conducted and the machine learning algorithms were applied to optimize the performance. Based on the experiment conducted, it produced an overall accuracy rate of 99.38 % using Sequential Minimal Optimization (SMO) algorithm. This performance criteria result indicated that this model is an efficient worm response model. � Springer International Publishing Switzerland 2015.

An efficient approximate method for solving Fredholm-Volterra integral equations of the third kind is presented. As a basis functions truncated Legendre series is used for unknown function and Gauss-Legendre quadrature formula with collocation method are applied to reduce problem into linear algebraic equations. The existence and uniqueness solution of the integral equation of the 3rd kind are shown as well as rate of convergence is obtained. Illustrative examples revels that the proposed method is very efficient and accurate. Finally, comparison results with the previous work are also given. � 2017, American Institute of Mathematical Sciences. All rights reserved.

In this note, we have developed new classes of one dimensional orthogonal polynomials Zk(i,n)(x),i = {1,2},n = 0,1,2,..., namely extended Chebyshev polynomials (ECPs) of the first and second kinds, which are an extension of the Chebyshev polynomials of the first and second kinds respectively. For non-homogeneous SIEs (bounded and unbounded case) truncated series of the first and second kind of ECPs are used to find approximate solution. It is found that first and second kinds of ECPs Zk(in)(x),i = {1,2} are orthogonal with weights w(1,k)(x)= xk-1/1-x2k and w(2,k)(x) = xk-11-x2k, where k is positive odd integer. Spectral properties of first and second kind of ECPs are also proved. Finally, two examples are presented to show the validity and accuracy of the proposed method.

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated. � 2013 Asma Ali Elbeleze et al.

The homotopy perturbation method, Sumudu transform, and He's polynomials are combined to obtain the solution of fractional Black-Scholes equation. The fractional derivative is considered in Caputo sense. Further, the same equation is solved by homotopy Laplace transform perturbation method. The results obtained by the two methods are in agreement. The approximate analytical solution of Black-Scholes is calculated in the form of a convergence power series with easily computable components. Some illustrative examples are presented to explain the efficiency and simplicity of the proposed method. © 2013 Asma Ali Elbeleze et al.

Internet Service Providers (ISPs) nowadays deal with high demand to promote good quality information. However, the knowledge to develop new pricing scheme that serve both customers and supplier is known, but only a few pricing plans involve QoS networks. This study will seek new proposed pricing plans are offered with QoS networks involved. The single link multi QoS Networks scheme is solved as an optimization model by comparing two models in multi QoS networks. The decisions whether to set up base price to be fixed to recover the cost or to be varied to compete in the market are considered. Also, the options of quality premium to be fixed to enable user to choose classes according to their preferences and budget or to be varied to enable ISP to promote certain service are set up. � 2013 Asian Network for Scientific Information.

This article will analyze new improved charging scheme with base price, quality premium and QoS networks involved. Sain and Herpers [5] already attempted to obtain revenue maximization by creating charging scheme of internet. The plan is attempted to solve multi service networks scheme as an optimization model to obtain revenue maximization using our improved model based on Byun and Chatterjee [2] and Sain and Herpers [5]. The results show that improved model can be solved optimally using optimization tool LINGO to achieve better revenue maximization. Better results are obtained in all cases rather than in [5]. The advantage of our new model is that ISP also can set up their base price and quality premium based on ISP preferences. For some cases for getting revenue maximization, we do not offer one service and just utilize some of the services.

In this paper, we propose a modification to homotopy perturbation method and improve to accelerate the rate of convergence in solving linear second-order Fredholm integro-differential equations. Some examples are given to show that this method is easy to apply and the results is obtained very fast. � 2016, University of Nis. All Rights Reserved.

In this note, we propose a new class of orthogonal polynomials (named Bachok–Hasham polynomials H1nk(x)), which is an extension of the Chebyshev polynomials. Eigenfunctions and corresponding eigenvalues are found for the homogeneous second kind of Logarithmic Singular Integral Equations (LogSIEs). For non-homogeneous LogSIEs truncated series of the first kind Bachok–Hasham polynomials are used to find approximate solution. It is found that first kind of Bachok–Hasham polynomials (H1nk(x)) are orthogonal with weight [Formula presented], where k is positive odd integer. Properties of first kind of Bachok–Hasham polynomials are also proved. Finally, two examples are presented to show the validity and accuracy of the proposed method. © 2019 THE AUTHORS

We apply the homotopy perturbation method to obtain the solution of partial differential equations of fractional order. This method is powerful tool to find exact and approximate solution of many linear and nonlinear partial differential equations of fractional order. Convergence of the method is proved and the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. The fractional derivatives are described in the Caputo sense. Some examples are presented to verify convergence hypothesis and simplicity of the method. � 2014 Asma Ali Elbeleze et al.

We implement relatively analytical methods, the homotopy perturbation method and the variational iteration method, for solving singular fractional partial differential equations of fractional order. The process of the methods which produce solutions in terms of convergent series is explained. The fractional derivatives are described in Caputo sense. Some examples are given to show the accurate and easily implemented of these methods even with the presence of singularities. � 2014 Asma Ali Elbeleze et al.

The research on cardinality of polynomials was started by Mohd Atan [1] when he considered a set, V(f-;p?)={umodp?:f-(u)?0modp?}, where ? > 0 and f-=(f1,f2,?fn). The term f-(u)?0-modp? means that we are considering all congruence equations of modulo p? and we are looking for u that makes the congruence equation equals zero. This is called the zeros of polynomials. The total numbers of such zeros is termed as N(f-;p?). The above p is a prime number and Zp is the ring of p-adic integers, and x-=(x1,x2, ?xn). He later let N(f-;p?)=card V(f-;p?). The notation N(f-;p?) means the number of zeros for that the polynomials f-. For a polynomial f (x) defined over the ring of integers Z, Sandor [2] showed that N(f;p?)? mp12ordpD, where D ? 0, ? > ordpD and D is the discriminant of f. In this paper we will try to introduce the concept of symbolic manipulation to ease the process of transformation from two-variables polynomials to one-variable polynomials. � 2016 Author(s).

Internet Service Providers (ISPs) nowadays deal with high demand to promote good quality information. However, the knowledge to develop new pricing scheme that serve both customers and supplier is known, but only a few pricing plans involve QoS networks. This study will seek new proposed pricing plans are offered with multi service multi link networks involved. The multi service multi link Networks scheme is solved as an optimization model by comparing two models in multi QoS networks. The results showed that by fixing the base price and varying the quality premium or varying the base price and quality and setting up the equal capacity link values, ISP achieved the goal to maximize the profit. � Springer International Publishing Switzerland 2015.