Implicit numerical technique for solving fractional Navier-Stoks equation with non-singular kernel

Recently, many authors have found analytical and numerical solutions of fractional Boussinesq equation by applying various fractional operators with singular kernels. Motivated by recently introduced fractional operators with non-singular kernels, in this paper a comparison of the solution of linearised fractional Boussinesq equation has been made for the fractional operators Caputo (with singular kernel) and Caputo-Fabrizio (with non-singular kernel). Linearised Boussinesq equation is derived by assuming that the average thickness of saturated layer of an aquifer is constant.

Two step Adams Bashforth method for time fractional Tricomi equation with non-local and non-singular Kernel

The paper aims to analyze the fractional Bloch equation with Caputo and Atangana–Baleanu–Caputo (ABC) derivative having a nonsingular kernel. The fixed point theorem is used to prove the existence and uniqueness of the proposed equation. Furthermore, the approximate‐analytical solution of the proposed equation is obtained by Aboodh transform iterative method (ATIM) in the form of a convergent series. The technique (ATIM) combines the new iterative method with the Aboodh transform. The convergence analysis of the approximate solution is discussed using the Cauchy convergence theorem. The validity of the ATIM is shown by numerical simulation and graphs. Also, the preference for nonsingular kernel over singular kernel is shown with the help of tables and graphs. For integer order, the obtained solutions from Caputo and ABC derivative are compared with the exact solutions and published work.

In this work, a general class of pantograph type nonlinear fractional integro-differential equations (PT-FIDEs) with non-singular and non-local kernel is considered. A numerical scheme based on the orthogonal basis functions including the shifted Legendre polynomials (SLPs) is proposed. First, we expand the unknown function and its derivatives in terms of the SLPs with unknown coefficients. Then, we present several theorems based on the SLPs for the help to achieve the approximate solution of the problem under study. Finally, by utilizing these theorems together with the collocation points, the main problem is transformed to a system of linear or nonlinear algebraic equations, which can be simply solved. An investigation for error estimate is discussed. The accuracy and efficiency of the proposed scheme are reported by four illustrative examples.

This paper investigates the existence, uniqueness, and monotonicity of positive solutions for a new class of hybrid nonlinear fractional integro-differential equations involving the Caputo–Fabrizio derivative. Unlike the classical Caputo operator, the Caputo–Fabrizio derivative is characterized by a non-singular exponential kernel, which leads to improved analytical tractability and better modeling of memory effects. Using Dhage's fixed point theorem, sufficient conditions for the well-posedness of the problem are derived. A concrete example is provided to demonstrate the effectiveness of the theoretical results, showing the relevance of the Caputo–Fabrizio approach in fractional modeling.

In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder’s and Banach’s fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.

<abstract><p>In this research, we investigate the direct interaction acquisition method to create a stochastic computational formula of cholera infection evolution via the fractional calculus theory. Susceptible people, infected individuals, medicated individuals, and restored individuals are all included in the framework. Besides that, we transformed the mathematical approach into a stochastic model since it neglected the randomization mechanism and external influences. The descriptive behaviours of systems are then investigated, including the global positivity of the solution, ergodicity and stationary distribution are carried out. Furthermore, the stochastic reproductive number for the system is determined while for the case $ \mathbb{R}_{0}^{s} > 1, $ some sufficient condition for the existence of stationary distribution is obtained. To test the complexity of the proposed scheme, various fractional derivative operators such as power law, exponential decay law and the generalized Mittag-Leffler kernel were used. We included a stochastic factor in every case and employed linear growth and Lipschitz criteria to illustrate the existence and uniqueness of solutions. So every case was numerically investigated, utilizing the newest numerical technique. According to simulation data, the main significant aspects of eradicating cholera infection from society are reduced interaction incidence, improved therapeutic rate, and hygiene facilities.</p></abstract>

In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

Fractional delay integrodifferential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

Spectral element technique for nonlinear fractional evolution equation, stability and convergence analysis

In this paper, a numerical technique by using fractional order Euler functions (FEFs) is presented to obtain the solution of Conformable fractional order Logistic equation. The operational matrix for Conformable fractional differentiation of FEFs is derived and the problem is reduced into a system of algebraic equations. Numerical examples are illustrated to express the efficiency of the proposed method.

This paper studied the fractional-order telegraph equations via the natural transform decomposition method with nonsingular kernel derivatives. The fractional result considered in the Caputo-Fabrizio derivative is Caputo sense. Currently, the communication system plays a vital role in a global society. High-frequency telecommunications continuously receive significant attention in the industry due to a slew of radiofrequency and microwave communication networks. These technologies use transmission media to move information-carrying signals from one location to another. We used natural transformation on fractional telegraph equations followed by inverse natural transformation to achieve the solution of the equation. To validate the technique, we have considered a few problems and compared them with the exact solutions.

In this paper, an efficient numerical scheme is constructed to approximate the solutions of the two-dimensional linear and nonlinear fractional cable equation. The fractional derivative is defined in the Atangana-Baleanu-Caputo sense. The nonstandard implicit compact finite difference method is presented to solve numerically the proposed equation. Special attention is given to studying the stability analysis of the numerical technique. Moreover, the truncation error is analyzed. Finally, two numerical test examples and comparative studies are presented. The numerical experiments demonstrate the high accuracy of the new method.

This paper adapts a new numerical technique for solving two- dimensional fractional integral equations with weakly singular. Using the spectral collocation method, the fractional operators of Legendre and Chebyshev polynomials, and Gauss-quadrature formula, we achieve a reduction of given problems into those of a system of algebraic equa tions. We apply the reported numerical method to solve several nu merical examples in order to test the accuracy and validity. Thus, the novel algorithm is more responsible for solving two-dimensional frac tional integral equations with weakly singular.

ABSTRACTIn this paper, an efficient and robust numerical technique is suggested to solve fractional Volterra integro-differential equations (FVIDEs). The proposed method is mainly based on the generalized fractional-order Legendre wavelets (GFLWs), their operational matrices and the Collocation method. The main advantage of the proposed method is that, by using the GFLWs basis, it can provide more efficient and accurate solution for FVIDEs in compare to integer-order wavelet basis. A comparison between the achieved results confirms accuracy and superiority of the proposed GFLWs method for solving FVIDEs. Error analysis and convergence of the GFLWs basis is provided.