Nonhomogeneous Fractional Differential Equations Research Articles

Solution to Fourier Transforms Method to Complex Variable of Non-homogeneous Fractional Differential Equations

This work is devoted to the study of fractional differential equations involving Caputo non-homogeneous fractional differential equations. Using Fourier transform method, a complex variable explicit solution to non-homogeneous second-order fractional differential equation was obtained.

A homotopy-based computational scheme for two-dimensional fractional cable equation

In this paper, we examine the time-dependent two-dimensional cable equation of fractional order in terms of the Caputo fractional derivative. This cable equation plays a vital role in diverse areas of electrophysiology and modeling neuronal dynamics. This paper conveys a precise semi-analytical method called the q-homotopy analysis transform method to solve the fractional cable equation. The proposed method is based on the conjunction of the q-homotopy analysis method and Laplace transform. We explained the uniqueness of the solution produced by the suggested method with the help of Banach’s fixed-point theory. The results obtained through the considered method are in the form of a series solution, and they converge rapidly. The obtained outcomes were in good agreement with the exact solution and are discussed through the 3D plots and graphs that express the physical representation of the considered equation. It shows that the proposed technique used here is reliable, well-organized and effective in analyzing the considered non-homogeneous fractional differential equations arising in various branches of science and engineering.

An Adomian decomposition method with some orthogonal polynomials to solve nonhomogeneous fractional differential equations (FDEs)

<p>The present study introduced modifications to the standard Adomian decomposition method (ADM) by combining the Taylor series with orthogonal polynomials, such as Legendre polynomials and the first and second kinds of Chebyshev polynomials. These improvements can be applied to solve fractional differential equations with initial-value problems in the Caputo sense. The approaches are based on the use of orthogonal polynomials, which are essential components in approximation theories. The study carefully analyzed their respective absolute error differences, highlighting the computational benefits of the proposed modifications, which offer improved accuracy and require fewer computational steps. The effectiveness and accuracy of the approach were validated through numerical examples, confirming its efficiency and reliability.</p>

Undetermined Coefficients Method for Sequential Fractional Differential Equations

The undetermined coefficients method is presented for nonhomogeneous sequential fractional differential equations involving Caputo fractional derivative of order n\alpha where n-1 n\alpha\le\ n and n\in\mathbb{N}. By employing proposed method, a particular solution of the considered equation is obtained. Some details about estimating the particular solution required to apply this method are explained. This method is shown to be particularly effective for nonhomogeneous fractional differential equations when the fractional differential equations involve some specific right-hand side functions.

IDENTITIES OF A GENERAL MULTIPLE HURWITZ-LERCH ZETA FUNCTION AND APPLICATIONS

In this article we introduce a general multiple Hurwitz-Lerch Zeta function. Then its convergence conditions and identities are obtained under certain conditions. We also derive some of connections to the multiple Hurwitz-Lerch Zeta function based upon Srivastava-Daoust hypergeometric series in several variables and other related functions of one and more variables found in the literature. Further, we study its integral representations and nd their applications for deriving generating relations and solving the non-homogeneous fractional differential equation.

Exact solutions and Hyers-Ulam stability of fractional equations with double delays

In this paper, we discuss the exact solutions of linear homogeneous and nonhomogeneous fractional differential equations with double delays. Firstly, a new concept of double-delayed Mittag-Leffler type matrix function is introduced, which is the promotion of the double-delayed matrix exponential. Secondly, we apply the double-delayed Mittag-Leffler type matrix function and Laplace transform approach to obtain the exact solutions of fractional differential equations with double delays. Furthermore, the solution is used to investigate the Hyers-Ulam stability of the system. Lastly, we illustrate our techniques by an example.

Numerical analysis of fractional differential equation by TSI-wavelet method

In this paper, we propose a new numerical algorithm for the approximate solution of non-homogeneous fractional differential equation. Using this algorithm the fractional differential equations are transformed into a system of algebraic linear equations by operational matrices of block-pulse and hybrid functions. Based on our new algorithm, this system of algebraic linear equations can be solved by a proposed (TSI) method. Further, some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm.

Well-Posedness and Approximation for Nonhomogeneous Fractional Differential Equations

In this paper, we consider the well-posedness and approximation for nonhomogeneous fractional differential equations in Banach spaces E. Firstly, we get the necessary and sufficient condition for the well-posedness of nonhomogeneous fractional Cauchy problems in the spaces Secondly, by using implicit difference scheme and explicit difference scheme, we deal with the full discretization of the solutions of nonhomogeneous fractional differential equations in time variables, get the stability of the schemes and the order of convergence.

Analysis of fractional duffing oscillator

In this contribution, a simple analytical method (which is an elegant combination of a well known methods; perturbation method and Laplace method) for solving non-linear and non-homogeneous fractional differential equations is pro- posed. In particular, the proposed method was used to analysed the fractional Duffing oscillator.The technique employed in this method can be used to analyse other nonlinear fractional differential equations, and can also be extended to non- linear partial fractional differential equations.The performance of this method is reliable, effective and gives more general solution.

Дифференциальные уравнения типа Эйлера дробного порядка

Дано решение неоднородного дифференциального уравнения типа Эйлера с дробными производными Римана – Лиувилля на полуоси в классе функций, представимых дробным интегралом, в терминах дробного аналога функции Грина. Построены дробные аналоги функции Грина в том случае, когда все корни характеристического многочлена различны, а также в случае, когда среди корней характеристического многочлена есть кратные. Сформулированы и доказаны теоремы разрешимости неоднородного дробно-дифференциального уравнения типа Эйлера на полуоси. Рассмотрены некоторые частные случаи и примеры.

Numerical solution of fractional differential equation by wavelets and hybrid functions

In this paper, we introduce methods based on operational matrix of fractional order integration for solving a typical n-term non-homogeneous fractional differential equation (FDE). We use Block pulse wavelets matrix of fractional order integration where a fractional derivative is defined in the Caputo sense. Also we consider Hybrid of Block-pulse functions and shifted Legendre polynomials to approximate functions. By uses these methods we translate an FDE to an algebraic linear equations which can be solve. Methods has been tested by some numerical examples.

Initial boundary value problems for a fractional differential equation with hyper-Bessel operator

Abstract Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart of a hyper-Bessel differential operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansions and results on existence and uniqueness are established. To solve the resultant equations, a solution to such kind of non-homogeneous fractional differential equation is also presented.

Analytical solutions of time-fractional models for homogeneous Gardner equation and non-homogeneous differential equations

In this paper, we obtain analytical solutions of homogeneous time-fractional Gardner equation and non-homogeneous time-fractional models (including Buck-master equation) using q-Homotopy Analysis Method (q-HAM). Our work displays the elegant nature of the application of q-HAM not only to solve homogeneous non-linear fractional differential equations but also to solve the non-homogeneous fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for non-linear differential equations. Comparisons are made upon the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.

On solving non-homogeneous fractional differential equations of Euler type

In this paper, we investigate non-homogeneous fractional differential equations of Euler type with alpha-left Riemann–Liouville fractional derivatives. In fact, these fractional differential equations are analogs of Euler ordinary differential equations. The classical power series method is used to obtain a useful formula to compute a particular solution of these equations. This formula is explicit and easy to compute by using Maple software or by setting a computer code to get explicit particular solution. The related results are proved for Euler ordinary differential equations.

The Harnack inequality for the Riemann-Liouville fractional derivation operator

In this note we establish the Harnack inequality for the Riemann-Liouville fractional derivation operator ∂ t of order α ∈ (0, 1). Here the function under consideration is assumed to be globally nonnegative. We show that the Harnack inequality in general fails if this global positivity assumption is replaced by a local one. A Harnack estimate is also derived for nonnegative solutions of a class of nonhomogeneous fractional differential equations. AMS subject classification: 26A33, 45D05, 47G20