A new operational matrix based on Bernoulli polynomials

In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized to transform the differential equation to a matrix equation which corresponds to a system of algebraic equations with unknown Bernoulli coefficients. This method can be used for many problems such as differential equations, integral equations and so on. Numerical examples show the method is computationally simple and also illustrate the efficiency and accuracy of the method

A technique for solving system of generalized Emden-Fowler equation using Legendre wavelet

This article is concerned with the development of an efficient numerical algorithm for the solution of a system of generalized nonlinear Emden-Fowler equation. The proposed algorithm is based on the Legendre wavelet operational matrix of integration technique. This method decreases the storage and computational complexity due to its calculation on the subinterval [ (n-1)/2^(k-1) , n/2^(k-1)) of [0,1]. The main highlight of this method is to converts the system of the differential equation into an equivalent system of nonlinear algebraic equations, which greatly simplifies approximation. Some numerical example shows that the proposed scheme is very efficient and reliable.Publisher's Versio

Boubaker Wavelets Functions: Properties and Applications

تم في هذا البحث تقديم شرح تفصيلي لدوال متعددة حدود بوبكر المتعامدة مع بعض الخواص ذات الاهمية، كذلك استنتاج تعريف متعددات حدود بوبكر الموجية في الفترة (1, 0] وذلك بالاستفادة  من بعض الخواص المهمة لمتعددة حدود بوبكر. تمتلك هذه الدوال الاساسية خاصية العيارية المتعامدة بالإضافة الى ضرورة تواجد المنطلق المرصوص. لهذه الدوال الموجية العديد من المزايا وقد استخدمت في المجال النظري بالإضافة الى المجال العملي وتم استخدامها مع متعددات الحدود المتعامدة لغرض طرح طريقة جديدة للتعامل مع العديد من المسائل في العلوم والهندسة ولذلك تعتبر طريقة استخدام الموجبات ذات اهمية كبيرة عند الاستفادة منها في المجالات ذات العلاقة. بالإضافة الى الاستفادة من موجبات بوبكر للحصول على خاصية جديدة  وهي مشتقات دالة بوبكر الموجية. استخدمت موجية بوبكر مع طريقة الترصيف للحصول على حل عددي تقريبي لمعادلات لان ايمدن من النوع الخطي المنفرد. تصف معادلات لان ايمدن العديد من الظواهر المهمة في علم الرياضيات والفيزياء السماوي مثل الانفجارات الحرارية الكونية وتكوين النجوم. وتعتبر احدى حالات مسائل القيمة الابتدائية المنفردة للمعادلات التفاضلية اللاخطية من الرتبة الثانية. تقوم هذه الطريقة المقترحة بتحويل معادلة لان ايمدن الى نظام من المعادلات التفاضلية الخطية والتي يمكن حلها بسهولة باستخدام الحاسبة. بناءً على هذا فقد ظهر تطابق الحل العددي مع الحل التحليلي بالرغم من استخدام عدد قليل من متعددات حدود بوبكر الموجية لغرض ايجاد هذا الحل. كذلك، تم في هذا البحث البرهنة على قيمه قيد الخطأ المستخرج من هذه الطريقة. وتضمن هذا البحث على ثلاث امثلة عددية من نوع معادلات لان ايمدن لتوضيح قابلية استخدام الطريقة المقترحة. تم توضيح النتائج الحقيقة مع النتائج التقريبية في شكل جداول ورسوم هندسية لغرض المقارنة.This paper is concerned with introducing an explicit expression for orthogonal Boubaker polynomial functions with some important properties. Taking advantage of the interesting properties of Boubaker polynomials, the definition of Boubaker wavelets on interval [0,1) is achieved. These basic functions are orthonormal and have compact support. Wavelets have many advantages and applications in the theoretical and applied fields, and they are applied with the orthogonal polynomials to propose a new method for treating several problems in sciences, and engineering that is wavelet method, which is computationally more attractive in the various fields. A novel property of Boubaker wavelet function derivative in terms of Boubaker wavelet themselves is also obtained. This Boubaker wavelet is utilized along with a collocation method to obtain an approximate numerical solution of singular linear type of Lane-Emden equations. Lane-Emden equations describe several important phenomena in mathematical science and astrophysics such as thermal explosions and stellar structure. It is one of the cases of singular initial value problem in the form of second order nonlinear ordinary differential equation. The suggested method converts Lane-Emden equation into a system of linear differential equations, which can be performed easily on computer. Consequently, the numerical solution concurs with the exact solution even with a small number of Boubaker wavelets used in estimation. An estimation of error bound for the present method is also proved in this work. Three examples of Lane-Emden type equations are included to demonstrate the applicability of the proposed method. The exact known solutions against the obtained approximate results are illustrated in figures for compariso

Advanced numerical scheme and its convergence analysis for a class of two-point singular boundary value problems.

[EN]In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation for approximating a class of two-point SBVPs with Dirichlet and Neumann–Robin boundary conditions is proposed. For the new approach, an initial guess is postulated in contrast to the boundary conditions in the first step. The second step deals with the usage of the Legendre wavelet operational matrix method to solve the initial value problem (IVP). Further, the resulting solution of the IVP is utilized at the second endpoint of the domain of a differential equation in a shooting projection method to improve the initial condition. These two steps are repeated until the desired accuracy of the solution is achieved. To support the mathematical formulation, a detailed convergence analysis of the new approach is conducted. The new approach is tested against some existing methods such as various types of the variational iteration method, considering several numerical examples to which it provides high-quality solutions

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

Vieta-Lucas Wavelet based schemes for the numerical solution of the singular models

In this paper, numerical methods based on Vieta-Lucas wavelets are proposed for solving a class of singular differential equations. The operational matrix of the derivative for Vieta-Lucas wavelets is derived. It is employed to reduce the differential equations into the system of algebraic equations by applying the ideas of the collocation scheme, Tau scheme, and Galerkin scheme respectively. Furthermore, the convergence analysis and error estimates for Vieta-Lucas wavelets are performed. In the numerical section, the comparative analysis is presented among the different versions of the proposed Vieta-Lucas wavelet methods, and the accuracy of the approaches is evaluated by computing the errors and comparing them to the existing findings.Comment: 23 pages, 4 figures, 2 Table

Accurate spectral solutions of first and second-order initial value problems by the ultraspherical wavelets-Gauss collocation method

In this paper, we present an ultraspherical wavelets-Gauss collocation method for obtaining direct solutions of first- and second-order nonlinear differential equations subject to homogenous and nonhomogeneous initial conditions. The properties of ultraspherical wavelets are used to reduce the differential equations with their initial conditions to systems of algebraic equations, which then must be solved by using suitable numerical solvers. The function approximations are spectral and have been chosen in such a way that make them easy to calculate the expansion coefficients of the thought-for solutions. Uniqueness and convergence of the proposed function approximation is discussed. Four illustrative numerical examples are considered and these results are comparing favorably with the analytic solutions and proving more accurate than those discussed by some other existing techniques in the literature