Solving Nonlinear Fractional Integro-Differential Equations of Volterra Type Using Novel Mathematical Matrices (original) (raw)
Journal of Function Spaces, 2021
This paper presents the nonlinear systems of Volterra-type fractional integro-differential equation solutions through a Chebyshev pseudospectral method. The proposed method is based on the Caputo fractional derivative. The results that we get show the accuracy and reliability of the present method. Different nonlinear systems have been solved; the solutions that we get are compared with other methods and the exact solution. Also, from the presented figures, it is easy to conclude that the CPM error converges quickly as compared to other methods. Comparing the exact solution and other techniques reveals that the Chebyshev pseudospectral method has a higher degree of accuracy and converges quickly towards the exact solution. Moreover, it is easy to implement the suggested method for solving fractional-order linear and nonlinear physical problems related to science and engineering.
Analytical Treatment of Volterra Integro-Differential Equations of Fractional Derivatives
Mathematical Researches, 2016
In this paper the solution of the Volterra integro-differential equations of fractional order is presented. The proposed method consists in constructing the functional series, sum of which determines the function giving the solution of considered problem. We derive conditions under which the solution series, constructed by the method is convergent. Some examples are presented to verify convergence, efficiency and simplicity of the method.
International Journal of Research, 2016
In this paper, the technique of modified Generalized Differential Transformation Method (GDTM) is used to solve a system of Non linear integro-differential equations with initial conditions. Moreover, a particular example has been discussed in three different cases to show reliability and the performance of the modified method. The fractional derivative is considered in the Caputo sense .The approximate solutions are calculated in the form of a convergent series, numerical results explain that this approach is trouble-free to put into practice and correct when applied to systems integro-differential equations.
In this paper, the technique of modified Generalized Differential Transformation Method (GDTM) is used to solve a system of Non linear integro-differential equations with initial conditions. Moreover, a particular example has been discussed in three different cases to show reliability and the performance of the modified method. The fractional derivative is considered in the Caputo sense .The approximate solutions are calculated in the form of a convergent series, numerical results explain that this approach is trouble-free to put into practice and correct when applied to systems integro-differential equations.
In this paper, a numerical approach is developed for solving initial value problem of linear fractional Volterra integro-differential equations. The approximate solution is substituted into the model equation and then collocated using shifted Chebyshev polynomial and Standard collocation points to obtain a system of linear algebraic equations, which is then solved by Newton-Rapson's method. Several numerical examples were solved to demonstrate the accuracy, reliability and efficiency of the method.
Journal of Nigerian Society of Physical Sciences, 2022
In this work, a collocation technique is used to determine the computational solution to fractional order Fredholm-Volterra integro-differential equations with boundary conditions using Caputo sense. We obtained the linear integral form of the problem and transformed it into a system of linear algebraic equations using standard collocation points. The matrix inversion approach is adopted to solve the algebraic equation and substituted it into the approximate solution. We established the uniqueness and convergence of the method and some modelled numerical examples are provided to demonstrate the method's correctness and efficiency. It is observed that the results obtained by our new method are accurate and performed better than the results obtained in the literature. The study will be useful to engineers and scientists. It is advantageous because it addresses the difficulty in tackling fractional order Fredholm-Volterra integro-differential problems using a simple collocation strategy. The approach has the advantage of being more accurate and reducing computer running time.
Acta Universitatis Apulensis, 2019
In this paper, Adomian decomposition and modified Laplace Ado-mian decomposition methods are successfully applied to find the approximate solution of Volterra integro-differential equation of fractional order. The reliability of the methods and reduction in the size of the computational work give these methods a wider applicability. Also, the behavior of the solution can be formally determined by analytical approximate. Moreover, we proved the convergence of the solutions. Finally, an example is included to demonstrate the validity and applicability of the proposed techniques.
Abstract and Applied Analysis, 2014
This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.
A Legendre collocation method for fractional integro-differential equations
Journal of Vibration and Control, 2011
A numerical method for solving the linear and non-linear fractional integro-differential equations of Volterra type is presented. The fractional derivative is described in the Caputo sense. The method is based upon Legendre approximations. The properties of Legendre polynomials together with the Gaussian integration method are utilized to reduce the fractional integro-differential equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results.
This paper demonstrates a study on some significant latest innovations in the approximation techniques to find the approximate solutions of Caputo fractional Volterra – Fredholm integro-differential equations. To apply this, the study uses Ado-mian decomposition method and modified Laplace Adomian decomposition method. A wider applicability of these techniques is based on their reliability and reduction in the size of the computational work. This study provides analytical approximate to determine the behavior of the solution. It proves the existence and uniqueness results and convergence of the solution. In addition, it brings an example to examine the validity and applicability of the proposed techniques.
2020
The main aim of this article is the extension of Optimal Homotopy Asymptotic Method to the system of fractional order integro-differential equations. The systems of fractional order Volterra integro-differential equations (SFIDEs) are taken as test examples. The fractional order derivatives are defined in the Caputo fractional form and the optimal values of auxiliary constants are calculated using the well-known method of least squares. The results obtained by proposed scheme are very encouraging and show close resemblance with exact values. Hence it will be more appealing for the researchers to apply the proposed scheme to different fractional order systems arising in different fields of sciences especially in fluid dynamics and bio-engineering. Introduction Fractional calculus has been concerned with integration and differentiation of fractional (non-integer) order of the function. In recent years, fractional calculus has been revolutionized by its tremendous innovations, observed in different fields of science and technology, such as fractional dynamics, nonlinear oscillation, hereditary in mechanics of solids, visco-elastically damped structures, bio-engineering and continuum mechanics [1-6]. Therefore, researchers have paid enormous interest in this field. Dynamical behavior of mixed type lump solution [7], exact optical solution of perturbed nonlinear Schrödinger equation [8], nonlinear complex fractional emerging telecommunication model [9], explicit solution of nonlinear Zoomeron equation[10], optical soliton in nematic liquid crystals [11], two-hybrid technique coupled with integral transformation for caputo time fractional Navier-Stokes equation [12], Analysis of Fractionally-Damped generalized Bagley-Torvik equation [13], Brusselator reaction-diffusion system [14], Fractional order of biological system [15], time fractional lvancevic option pricing model [16], analysis of fractionally damped beams [17], model of vibration equation of large membranes [18], fractional Jeffrey fluid over inclined plane [19], thermal stratification of rotational second-grade fluid [20], long memory processes [21], heat-transfer properties of noble gases [22] and modelling the dynamic mechanical analysis [23]. A history of fractional differential operators can be found in [24]. Owing to its applications , researchers compel to extract its solutions, but exact solutions of all problems are difficult to find due to its nonlinearity. Therefore, researchers used analytical and numerical techniques for its approximate solutions. Numerical methods [25-28], perturbation methods [29-31], homotopy based method [32,33] and iterative techniques [34,35] are the main tools for obtaining the approximation of nonlinear problems. In the literature, researchers have used different techniques for the solution of fractional order integro-differential equations and their systems. Khan et al. implemented the Chebyshev wavelet method [36], Rahim et al. used the fractional alternative Leg-endre functions [37], Hamoud et al. applied the modified adomian decomposition method [38], Zedan et al. used the Chebyshev spectral method [39], and Zada et al. studied the impulsive coupled system [40]. The OHAM was introduced by Marinca et al. [41-43] for the solution of differential equations, and in a short period, different researchers have successfully implemented it for the solution of different problems
SOLVING FRACTIONAL VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS BY USING ALTERNATIVE LEGENDRE FUNCTIONS
Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms , 2021
This paper mainly focuses on numerical technique based on a new set of functions called fractional alternative Legendre for solving the Volterra integro-differential equations of fractional order. Also, the convergence analysis of the proposed method is investigated. Finally, some examples are included to demonstrate the validity and applicability of the proposed technique.
On the numerical solution of nonlinear fractional-integro differential equations
New Trends in Mathematical Science, 2017
In the present study, a numerical method, perturbation-iteration algorithm (shortly PIA), have been employed to give approximate solutions of nonlinear fractional-integro differential equations (FIDEs). Comparing with the exact solution, the PIA produces reliable and accurate results for FIDEs.
Alexandria Engineering Journal, 2022
The point of this paper is to analyze and investigate the analytic-approximate solutions for fractional system of Volterra integro-differential equations in framework of Caputo-Fabrizio operator. The methodology relies on creating the reproducing kernel functions to gain analytical solutions in a uniform form of a rapidly convergent series in the Hilbert space. Using the Gram-Schmidt orthonomalization process, the orthonormal basis system is constructed in a dense compact domain to encompass the Fourier series expansion in view of reproducing kernel properties. Besides, convergence and error analysis of the proposed technique are discussed. For this purpose, several numerical examples are tested to demonstrate the great feasibility and efficiency of the present method and to support theoretical aspect as well. From a numerical point of view, the acquired solutions simulation indicates that the methodology used is sound, straightforward, and appropriate to deal with many physical issues in light of Caputo-Fabrizio derivatives.
Mathematical sciences, 2020
In this paper, we present a numerical method to solve fractional-order delay integro-differential equations. We use the operational matrices based on the fractional-order Euler polynomials to obtain numerical solution of the considered equations. By approximating the unknown function and its derivative in terms of the fractional-order Euler polynomials and substituting these approximations into the original equation, the original equation is reduced to a system of nonlinear algebraic equations. The convergence analysis of the proposed method is discussed. Finally, some examples are included to show the accuracy and validity of the proposed method.
Dynamical Analysis of Fractional Integro-Differential Equations
Mathematics
In this article, we solve fractional Integro differential equations (FIDEs) through a well-known technique known as the Chebyshev Pseudospectral method. In the Caputo manner, the fractional derivative is taken. The main advantage of the proposed technique is that it reduces such types of equations to linear or nonlinear algebraic equations. The acquired results demonstrate the accuracy and reliability of the current approach. The results are compared to those obtained by other approaches and the exact solution. Three test problems were used to demonstrate the effectiveness of the proposed technique. For different fractional orders, the results of the proposed technique are plotted. Plotting absolute error figures and comparing results to some existing solutions reveals the accuracy of the proposed technique. The comparison with the exact solution, hybrid Legendre polynomials, and block-pulse functions approach, Reproducing Kernel Hilbert Space method, Haar wavelet method, and Pseudo...
A solution method for integro-differential equations of conformable fractional derivative
Thermal Science
The aim of this work is to determine an approximate solution of a fractional order Volterra-Fredholm integro-differential equation using by the Sinc-collocation method. Conformable derivative is considered for the fractional derivatives. Some numerical examples having exact solutions are approximately solved. The comparisons of the exact and the approximate solutions of the examples are presented both in tables and graphical forms.