Numerical approximations for population growth model by Rational Chebyshev and Hermite Functions collocation approach: A comparison

This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve the Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro-differential equations.Comment: 18 pages, 5 figures; Published online in the journal of "Mathematical Methods in the Applied Sciences

Shifted Jacobi spectral collocation method with convergence analysis for solving integro-differential equations and system of integro-differential equations

This article addresses the solution of multi-dimensional integro-differential equations (IDEs) by means of the spectral collocation method and taking the advantage of the properties of shifted Jacobi polynomials. The applicability and accuracy of the present technique have been examined by the given numerical examples in this paper. By means of these numerical examples, we ensure that the present technique is simple and very accurate. Furthermore, an error analysis is performed to verify the correctness and feasibility of the proposed method when solving IDE

Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

في هذا البحث، ستراتيجيات جديدة لإيجاد الحل العددي للمعادلات الخطية الكسورية التفاضلية - التكاملية فولتيرا- فريدهولم (LFVFIDE) تم دراستها. الطرق المتبعه على ثلاث انواع من متعددات الحدود لاكرانج وهي: متعددة حدود لاكرانج الأصلية (OLP) ، متعددة حدود لاكرانج ذات الدعامة المركزية (BLP) و متعددة حدود لاكرانج المعدلة  (MLP).كما تم اقتراح خوارزمية عامة واعطاء  أمثلة لبرهنة فعالية الطرق وتنفيذها. وأخيرًا ، تم استخدام مقارنة بين الطرق المقترحة والطرق الأخرى لحل هذا النوع من المعادلات.In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal method in solving these problems

On the Solution of Volterra Integro-differential Equations using a Modified Adomian Decomposition Method

The Adomian decomposition method’s effectiveness has been demonstrated in recent research, the process requires several iterations and can be time-consuming. By breaking down the source term function into series, the current work introduced a new decomposition approach to the Adomian decomposition method. As compared to the conventional Adomian decomposition approach, the newly devised method hastens the convergence of the solution. Numerical experiments were provided to show the superiority qualities

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples. &nbsp