Eman S. Al-Aidarous

Design of stochastic solvers based on genetic algorithms for solving nonlinear equations

Abstract In the present study, a novel intelligent computing approach is developed for solving nonlinear equations using evolutionary computational technique mainly based on variants of genetic algorithms (GA). The mathematical model of the equation is formulated by defining an error function. Optimization of fitness function is carried out with the competency of GA used as a tool for viable global search methodology. Comprehensive numerical experimentation has been performed on number of benchmark nonlinear algebraic and transcendental equations to validate the accuracy, convergence and robustness of the designed scheme. Comparative studies have also been made with available standard solution to establish the correctness of the proposed scheme. Reliability and effectiveness of the design approaches are validated based on results of statistical parameters.

Design of bio-inspired heuristic technique integrated with interior-point algorithm to analyze the dynamics of heartbeat model

Display Omitted Design of biological inspired heuristics to analyze the dynamics of heartbeat model.The strength of ANNs, GA, and IPAs is exploited to solve the Heartbeat dynamic system.Design scheme is tested effectively on variants of problems by taking different values of parameters in the system.Comparison from reference solution established the correctness of the proposed scheme.Results of performance indices validate consistent accuracy and convergence of the scheme. In this study, bio-inspired computing is presented for finding an approximate solution of governing system represents the dynamics of the HeartBeat Model (HBM) using feed-forward Artificial Neural Networks (ANNs), optimized with Genetic Algorithms (GAs) hybridized with Interiort-Point Algorithm (IPA). The modeling of the system is performed with ANNs by defining an unsupervised error function and optimization of unknown weights are carried out with GA-IPA; in which, GAs is used as an effective global search method and IPA for rapid local convergence. Design scheme is applied to study the dynamics of HBM by taking different values for perturbation factor, tension factor in the muscle fiber and the length of the muscle fiber in the diastolic state. A large number of simulations are performed for the proposed scheme to determine its effectiveness and reliability through different performance indices based on mean absolute deviation, Nash-Sutcliffe efficiency, and Thiels inequality coefficient.

A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions

We consider the ergodic (or additive eigenvalue) problem for the Neumann type boundary value problem for Hamilton-Jacobi equations and the corresponding discounted problems. When denoting by u the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family {u}λ>0 to a solution of the ergodic problem, as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting byproduct, we introduce Mather measures associated with Hamilton-Jacobi equations with the Neumann type boundary condtitions. These results are variants of the main results in the paper “Convergence of the solutions of the discounted equations” by A. Davini, A. Fathi, R. Iturriaga and M. Zavidovique, where they study the same convergence problem on smooth compact manifolds without boundary. Acknowledgments. The third author is grateful to Dr. Andrea Davini for sending him the preprint [6] at a timely occasion. This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, under grant no. 2-130-1433/HiCi. The authors, therefore, acknowledge with thanks DSR technical and financial support. The work of the third author was supported in part by KAKENHI #21340032, #21224001, #23340028 and #23244015, JSPS.

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {Tn(f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {Tn−1(g)Tn(f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

An efficient method for the static deflection analysis of an infinite beam on a nonlinear elastic foundation of one-way spring model

An efficient numerical iterative method is constructed for the static deflection of an infinite beam on a nonlinear elastic foundation. The proposed iterative scheme consists of quasilinear method (QLM) and Green’s function technique. The QLM translates the nonlinear ordinary differential equation into iterative linear ordinary differential equation. The successive iterations of quasilinear form of ordinary differential equation (ODE) show the quadratic convergence if an initial guess is chosen in the neighbourhood of true solution. The Green’s function technique converts the differential operator into an integral operator and the integral operator is approximated by discrete summation which finally gives us an iterative formula for the resulting set of algebraic equations.The numerical validity and efficiency are proved by simulating some nonlinear problems.

Existence and Stability of Solutions for Hadamard-Stieltjes Fractional Integral Equations

We give some existence results and Ulam stability results for a class of Hadamard-Stieltjes integral equations. We present two results: the first one is an existence result based on Schauder’s fixed point theorem and the second one is about the generalized Ulam-Hyers-Rassias stability.

An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions

One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.

Construction of a convergent scheme for finding matrix sign function

In this paper, a new method is constructed for finding matrix sign function. It is proven that it possesses the high convergence order nine with global behavior. Numerical experiments are also provided to support the theoretical discussions.

Higher order multi-step iterative method for computing the numerical solution of systems of nonlinear equations

In the present study, we consider multi-step iterative method to solve systems of nonlinear equations. Since the Jacobian evaluation and its inversion are expensive, in order to achieve a better computational efficiency, we compute Jacobian and its inverse only once in a single cycle of the proposed multi-step iterative method. Actually the involved systems of linear equations are solved by employing the LU-decomposition, rather than inversion. The primitive iterative method (termed base method) has convergence-order (CO) five and then we describe a matrix polynomial of degree two to design a multi-step method. Each inclusion of single step in the base method will increase the convergence-order by three. The general expression for CO is 3 s - 1 , where s is the number of steps of the multi-step iterative method. Computational efficiency is also addressed in comparison with other existing methods. The claimed convergence-rates proofs are established. The new contribution in this article relies essentially in the increment of CO by three for each added step, with a comparable computational cost in comparison with existing multi-steps iterative methods. Numerical assessments are made which justify the theoretical results: in particular, some systems of nonlinear equations associated with the numerical approximation of partial differential equations (PDEs) and ordinary differential equations (ODEs) are built up and solved.

Haar Wavelet Method for the System of Integral Equations

We employed the Haar wavelet method to find numerical solution of the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs). Five test problems, for which the exact solution is known, are considered. Comparison of the results is obtained by the Haar wavelet method with the exact solution.