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Showing papers in "International Journal of Computer Mathematics in 2006"


Journal ArticleDOI
TL;DR: Several computational techniques based on finite-difference schemes and the product trapezoidal numerical integration rule are constructed for determining the solution of a partial integro-differential equation subject to an initial condition and given boundary conditions.
Abstract: Partial integro-differential equations occur in many fields of science and engineering. This study presents various numerical schemes for solving a partial integro-differential equation with a weakly singular kernel. These schemes are presented for determining the solution of a partial integro-differential equation subject to an initial condition and given boundary conditions. We construct several computational techniques based on finite-difference schemes and the product trapezoidal numerical integration rule. This problem can be found in the modelling of physical phenomena involving viscoelasticity forces. The finite-difference procedures developed are based on the forward Euler explicit scheme, the backward Euler implicit technique, the Crank–Nicolson implicit formula and Crandall’s implicit method. Three of the methods have second-order accuracy with respect to the space variable. The order of accuracy of the Crandall’s scheme is higher than that of the others. The numerical results of a test problem ...

107 citations


Journal ArticleDOI
TL;DR: The sine–cosine method is used to construct exact periodic and soliton solutions of coupled nonlinear evolution equations and many new families of exact travelling wave solutions of the Konopelchenko–Dubrovsky equations and the coupled non linear Klein–Gordon and Nizhnik–Novikov–Veselov equations are successfully obtained.
Abstract: In this paper, we establish exact solutions for coupled nonlinear evolution equations. The sine–cosine method is used to construct exact periodic and soliton solutions of coupled nonlinear evolution equations. Many new families of exact travelling wave solutions of the (2+1)-dimensional Konopelchenko–Dubrovsky equations and the coupled nonlinear Klein–Gordon and Nizhnik–Novikov–Veselov equations are successfully obtained. The obtained solutions include compactons, solitons, solitary patterns and periodic solutions. These solutions may be important and of significance for the explanation of some practical physical problems.

98 citations


Journal ArticleDOI
TL;DR: A computational method using Haar wavelets to determine the piecewise constant feedback controls for a finite-time linear optimal control problem of a time-varying state-delayed system is presented.
Abstract: We present a computational method using Haar wavelets to determine the piecewise constant feedback controls for a finite-time linear optimal control problem of a time-varying state-delayed system. The method is simple and computationally advantageous. The approximate optimal trajectory and optimal control are calculated using the Haar wavelet integral, product, and delay operational matrices. An illustrative example is included to demonstrate the validity and applicability of the technique.

70 citations


Journal ArticleDOI
TL;DR: The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements with linear stability analysis.
Abstract: The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.

48 citations


Journal ArticleDOI
TL;DR: The three methods are compared and it is shown that the VIM is more efficient and effective than the ADM and the DTM, and also converges to its exact solution more rapidly.
Abstract: The implementation of the two-dimensional differential transform method (DTM), Adomian's decomposition method (ADM), and the variational iteration method (VIM) in the mathematical applications of partial differential equations is examined in this paper. The VIM has been found to be particularly valuable as a tool for the solution of differential equations in engineering, science, and applied mathematics. The three methods are compared and it is shown that the VIM is more efficient and effective than the ADM and the DTM, and also converges to its exact solution more rapidly. Numerical solutions of two examples are calculated and the results are presented in tables and figures.

44 citations


Journal ArticleDOI
TL;DR: Two operations in P systems capable of constructing an exponential number of membranes in linear time are studied and it is shown that in the framework of P systems with active membranes but without polarizations and in the context of P system with membrane creation, dissolution rules play a crucial role from the computational efficiency point of view.
Abstract: Trading (in polynomial time) space for time in the framework of membrane systems is not sufficient to efficiently solve computationally hard problems. On the one hand, an exponential number of objects generated in polynomial time is not sufficient to solve NP-complete problems in polynomial time. On the other hand, when an exponential number of membranes is created and used as workspace, the situation is very different. Two operations in P systems (membrane division and membrane creation) capable of constructing an exponential number of membranes in linear time are studied in this paper. NP-complete problems can be solved in polynomial time using P systems with active membranes and with polarizations, but when electrical charges are not used, then dissolution rules turn out to be very important. We show that in the framework of P systems with active membranes but without polarizations and in the framework of P systems with membrane creation, dissolution rules play a crucial role from the computational eff...

40 citations


Journal ArticleDOI
TL;DR: Daubechies' compactly supported wavelet basis has been used in this study and the results obtained are highly encouraging and can be computed for a large value of the linear growth rate.
Abstract: Fisher's equation, which describes the logistic growth–diffusion process and occurs in many biological and chemical processes, has been studied numerically by the wavelet Galerkin method. Wavelets are functions which can provide local finer details. The solution of Fisher's equation has a compact support property and therefore Daubechies' compactly supported wavelet basis has been used in this study. The results obtained by the present method are highly encouraging and can be computed for a large value of the linear growth rate.

34 citations


Journal ArticleDOI
TL;DR: Two new one-parameter families of methods for finding simple and real roots of non-linear equations without employing derivatives of any order are developed.
Abstract: Two new one-parameter families of methods for finding simple and real roots of non-linear equations without employing derivatives of any order are developed. Error analysis providing the fourth-order convergence is given. Each member of the families requires three evaluations of function per step, and therefore the method has an efficiency index of 1.587. Numerical examples are presented and the performance of the method presented here is compared with methods available in the literature.

33 citations


Journal ArticleDOI
TL;DR: A Taylor collocation method is presented for numerically solving the system of high-order linear Fredholm–Volterra integro-differential equations in terms of Taylor polynomials, valid for the systems of differential and integral equations.
Abstract: A Taylor collocation method is presented for numerically solving the system of high-order linear Fredholm–Volterra integro-differential equations in terms of Taylor polynomials. Using the Taylor collocations points, the method transforms the system of linear integro-differential equations (IDEs) and the given conditions into a matrix equation in the unknown Taylor coefficients. The Taylor coefficients can be found easily, and hence the Taylor polynomial approach can be applied. This method is also valid for the systems of differential and integral equations. Numerical examples are presented to illusturate the accuracy of the method. The symbolic algebra program Maple is used to prove the results.

32 citations


Journal ArticleDOI
TL;DR: A heuristic approach using an intercept matrix to identify redundant constraints prior to the start of the solution process and the tendency of variables to pop in and pop out of the basis is eradicated after eliminating the redundancies.
Abstract: Linear programming (LP) is one of the most important techniques used in modelling and solving practical optimization problems that arise in industry, commerce, and management When formulating an LP model, systems analysts and researchers often include all possible constraints although some of them may not be binding at the optimal solution The presence of redundant constraints does not alter the optimum solution(s), but may consume extra computational effort Many researchers have proposed algorithms for identifying the redundant constraints in LP models Here we propose a heuristic approach using an intercept matrix to identify redundant constraints prior to the start of the solution process An interesting observation of the proposal technique is that the tendency of variables to pop in and pop out of the basis is eradicated after eliminating the redundancies The eradication of pop-in and pop-out substantially reduces the number of iterations A significant reduction in the computational effort is ac

31 citations


Journal ArticleDOI
TL;DR: In this article, a new intensity-based similarity metric was proposed for the registration of multimodal images, which combines the robust estimation with both the forward and inverse transformation to reduce the negative effects of outliers in the images.
Abstract: This paper proposes a new intensity-based similarity metric that can be used for the registration of multimodal images. It combines the robust estimation with both the forward and inverse transformation to reduce the negative effects of outliers in the images. For this purpose, we firstly employ the multiresolution technique to downsample the original images, then resort to the simulated annealing method to initialize the transformation parameters at the coarsest resolution. Finally the Powell method is utilized to obtain the optimal transformation parameters at each resolution. In our experiments, the new method is compared to other popular similarity measures, on the synthetic data as well as the real data, and the experimental results are encouraging.

Journal ArticleDOI
TL;DR: The structure of the roots of the q-polynomials K n, q (x) for values of the index n using a computer is described.
Abstract: In this paper we explore the shapes of the q-numbers K n, q and the q-polynomials K n, q (x). Finally, we describe the structure of the roots of the q-polynomials K n, q (x) for values of the index n using a computer.

Journal ArticleDOI
TL;DR: It is shown that the proposed approach for significant rare data by introducing second support in discovering the association rules of such data provides better performance as compared to standard association rules techniques.
Abstract: Association rule is one of the data mining techniques involved in discovering information that represents the association among data. Data in the database sometimes appear infrequent but highly associated with a specific data. This paper proposes a technique for significant rare data by introducing second support in discovering the association rules of such data. We show that the proposed approach provides better performance as compared to standard association rules techniques.

Journal ArticleDOI
TL;DR: A numerical technique for solving a second-order nonlinear Neumann problem is presented based on semi-orthogonal B-spline wavelets and results show the efficiency of the proposed technique for the studied problem.
Abstract: A numerical technique for solving a second-order nonlinear Neumann problem is presented. The authors’ approach is based on semi-orthogonal B-spline wavelets. Two test problems are presented and numerical results are tabulated to show the efficiency of the proposed technique for the studied problem.

Journal ArticleDOI
TL;DR: It is proved that the semi-implicit Euler method is convergent with strong order p=0.5 and under which the method is asymptotic mean square stable.
Abstract: This paper deals with the convergence and stability of the semi-implicit Euler method for linear stochastic delay integro-differential equations. It is proved that the semi-implicit Euler method is convergent with strong order p=0.5. The condition under which the method is asymptotic mean square stable is determined and numerical experiments are presented.

Journal ArticleDOI
TL;DR: A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed, which is third-order accurate in time and O(2−jp ) accurate in space.
Abstract: A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed. The scheme is third-order accurate in time and O(2−jp ) accurate in space. Unlike Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving non-linear problems. The compactly supported orthogonal wavelet bases D6 developed by Daubechies are used in the Galerkin scheme. The proposed scheme is tested with both parabolic and hyperbolic partial differential equations. The numerical results indicate the versatility and effectiveness of the proposed scheme.

Journal ArticleDOI
TL;DR: A comparison between the Adomian decomposition method and the fourth-order Runge–Kutta method for solving the one-phase Stefan problem is presented.
Abstract: The solution of the one-phase Stefan problem is presented. A Stefan's task is first approximated with a system of ordinary differential equations. A comparison between the Adomian decomposition method and the fourth-order Runge–Kutta method for solving this system is then presented.

Journal ArticleDOI
TL;DR: This study deals with two drawbacks in a customary model for distinguishing the dynamically efficient paths in dynamic performance analysis and provides some corrections.
Abstract: This study deals with two drawbacks in a customary model for distinguishing the dynamically efficient paths in dynamic performance analysis and provides some corrections.

Journal ArticleDOI
Zhi-Hao Cao1
TL;DR: The properties of the corresponding preconditioned matrices, in particular the spectrum, are analysed, and the computational performance of preconditionsed iterative methods are discussed.
Abstract: We study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two-by-two block systems with a zero (2,2)-block. These block diagonal preconditioners are obtained by a sign change of the (2,2)-block of the block diagonal preconditioners constructed previously [De Sturler, E. and Liesen, J., 2005, Block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems. Part one: Theory. SIAM Journal on Scientific Computing, 26, 1598–1619]. We analyse the properties of the corresponding preconditioned matrices, in particular the spectrum, and discuss the computational performance of preconditioned iterative methods. Numerical experiments of a model Navier–Stokes problem are presented.

Journal ArticleDOI
TL;DR: Based on the semi-explicit asymmetric exponential schemes constructed by the author, a new alternating group explicit (AGE) method with exponential-type for the numerical solution of the convection–diffusion equation is derived.
Abstract: Based on the semi-explicit asymmetric exponential schemes constructed by the author, a new alternating group explicit (AGE) method with exponential-type for the numerical solution of the convection–diffusion equation is derived in the paper. The method has the obvious property of parallelism and is unconditionally stable. The results of numerical examples are given to show the effectiveness of the present methods that are in preference to the Evans and Abdullah' method in [Evans, D.J. and Abdullah, A.R., 1985, A new explicit method for the diffusion–convection equation. Computers & Mathematics with Application, 11, 145–154].

Journal ArticleDOI
TL;DR: The use of two constructed polynomial spline functions to approximate the solution of a system of first-order delay differential equations is described.
Abstract: The use of two constructed polynomial spline functions to approximate the solution of a system of first-order delay differential equations is described. The first spline function is a polynomial with an undetermined constant coefficient in the last term. The other has a polynomial spline form. The error analysis and stability of the second function are theoretically investigated and a test example is given. A comparison of the two forms is carried out to illustrate the pertinent features of the proposed techniques.

Journal ArticleDOI
TL;DR: It is shown that the intervals of stability are strongly stable within an appropriate region of stability and the method is thus suitable for oscillatory problems by applying the method to the test equation y″=−ω2 y, ω>0.
Abstract: An embedded diagonally implicit Runge–Kutta Nystrom (RKN) method is constructed for the integration of initial-value problems for second-order ordinary differential equations possessing oscillatory solutions. This embedded method is derived using a three-stage diagonally implicit RKN method of order four within which a third-order three stage diagonally implicit RKN method is embedded. We demonstrate how this system can be solved, and by an appropriate choice of free parameters, we obtain an optimized RKN(4,3) embedded algorithm. We also examine the intervals of stability and show that the method is strongly stable within an appropriate region of stability and is thus suitable for oscillatory problems by applying the method to the test equation y″=−ω2 y, ω>0. Necessary and sufficient conditions are given for this method to possess non-vanishing intervals of periodicity, for the fourth-order method. Finally, we present the coefficients of the method optimized for small truncation errors. This new scheme is...

Journal ArticleDOI
TL;DR: The parameters governing the arm model of a second-order robot control problem are studied using the Runge–Kutta-Butcher (RK–Butcher) algorithm to compare the corresponding discrete solutions at different time intervals.
Abstract: The parameters governing the arm model of a second-order robot control problem are studied using the Runge–Kutta–Butcher (RK–Butcher) algorithm. The exact solution of the system of equations representing the arm model of a robot are compared with the corresponding discrete solutions (approximate solutions) at different time intervals using the RK–Butcher algorithm. The absolute error between the exact and discrete solutions is also determined. A detailed well-composed comparison is carried out using the results and graphs obtained.

Journal ArticleDOI
TL;DR: A class of definite integrals are evaluated using the Cauchy residue theorem and used to obtain closed-form expressions for several infinite series associated with the Riemann zeta function.
Abstract: A class of definite integrals are evaluated using the Cauchy residue theorem. These evaluations are then used to obtain closed-form expressions for several infinite series associated with the Riemann zeta function. A number of interesting (known or new) special cases and consequences of the main results are also considered.

Journal ArticleDOI
TL;DR: It is proved that P colonies with only one agent working in the sequential mode are already computationally complete, and the family of Parikh sets generated by matrix grammars without appearance checking is characterized by P colony with prescribed teams consisting of only (at most) two sets containing exactly one evolution rule or one antiport rule whenWorking in the modes *,=1,≥1, and ≤ k for k ≥1.
Abstract: We further investigate the computational power of P colonies working in the maximally parallel and sequential modes. It turns out that there is a trade-off between maximal parallelism and checking programs: using checking programs (i.e. priorities on the communication rules in the programs of the agents), P colonies working in the sequential mode with a height of at most five are computationally complete, whereas when working in the maximally parallel mode, P colonies (again with height five) already obtain the same computational power without using checking programs. Moreover, when allowing an arbitrary number of programs for each agent, we can prove that P colonies with only one agent (thus these P colonies are working in the sequential mode) are already computationally complete. On the other hand, P colonies with an arbitrary number of agents working in the sequential mode, as well as even P colonies with only one agent using an arbitrary number of non-checking programs, characterize the family of Pari...

Journal ArticleDOI
TL;DR: A linearized implicit finite-difference method is presented to find numerical solutions of the equal width wave equation and has been used successfully to investigate the motion of a single solitary wave, the development of the interaction of two solitary waves and an undular bore.
Abstract: A linearized implicit finite-difference method is presented to find numerical solutions of the equal width wave equation. The method has been used successfully to investigate the motion of a single solitary wave, the development of the interaction of two solitary waves and an undular bore. The obtained results are compared with other numerical results in the literature. A stability analysis of the scheme is also investigated.

Journal ArticleDOI
TL;DR: The bifurcation method of the dynamical and numerical approach to differential equations to study higher order wave equations of Korteweg–de Vries (KdV) type is used and the results show that the numerical integrations are identical with the theoretical derivations.
Abstract: In this paper, the bifurcation method of the dynamical and numerical approach to differential equations to study higher order wave equations of Korteweg–de Vries (KdV) type is used. With this methodology we obtain the compacton-like and kink-like wave solutions of the high order KdV type equation. Their implicit expressions are given and their planar graphs are simulated. The results show that the numerical integrations are identical with the theoretical derivations.

Journal ArticleDOI
TL;DR: A solution procedure consisting of fuzzy goal programming and stochastic simulation-based genetic algorithm is presented to solve multiobjective chance constrained programming problems with continuous random variables in the objective functions and in chance constraints.
Abstract: Solution procedure consisting of fuzzy goal programming and stochastic simulation-based genetic algorithm is presented, in this article, to solve multiobjective chance constrained programming problems with continuous random variables in the objective functions and in chance constraints. The fuzzy goal programming formulation of the problem is developed first using the stochastic simulation-based genetic algorithm. Without deriving the deterministic equivalent, chance constraints are used within the genetic process and their feasibilities are checked by the stochastic simulation technique. The problem is then reduced to an ordinary chance constrained programming problem. Again using the stochastic simulation-based genetic algorithm, the highest membership value of each of the membership goal is achieved and thereby the most satisfactory solution is obtained. The proposed procedure is illustrated by a numerical example.

Journal ArticleDOI
TL;DR: This paper derives the distributions of some pivotal quantities that can be used to predict future order statistics from exponential lifetime when the sample size is Poisson and binomial distributed and calculates the percentage points (factors) of the predictive distributions and uses them to construct predictive confidence intervals for the future order Statistics.
Abstract: In this paper, we generalize the work by Lawless (1971) and Lingappaiah (1973) for predicting future order statistics from exponential lifetime when the sample size is a random variable. First, we derive the distributions of some pivotal quantities that can be used to predict future order statistics from exponential lifetime when the sample size is Poisson and binomial distributed. Next, we calculate the percentage points (factors) of the predictive distributions and use them to construct predictive confidence intervals for the future order statistics. To show the usefulness of our results, we present some simulation experiments. Finally, we apply our techniques to some real data sets in life testing.

Journal ArticleDOI
TL;DR: It is shown that P systems with restricted membrane creation (the newly created membrane can only be of the same kind as the parent one) generate at least matrix languages, even when having at most one object in the configuration (except the environment).
Abstract: We improve, by using register machines, some existing universality results for specific models of P systems. P systems with membrane creation are known to generate all recursively enumerable sets of vectors of non-negative integers, even when no region (except the environment) contains more than one object of the same kind. We show here that they generate all recursively enumerable languages, and that two membrane labels are sufficient (the same result holds for accepting all recursively enumerable vectors of non-negative integers). Moreover, at most two objects are present inside the system at any time in the generative case. We then prove that 10+m symbols are sufficient to generate any recursively enumerable language over m symbols. P systems with active membranes without polarizations are known to generate all recursively enumerable sets of vectors of non-negative integers. We show that they generate all recursively enumerable languages; four starting membranes with three labels or seven starting memb...