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This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville's fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method's effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques. [ABSTRACT FROM AUTHOR]

The Caputo fractional α -derivative, 0 < α < 1 , for non-smooth functions with 1 + α regularity is calculated by numerical computation. Let I be an interval and D α (I) be the set of all functions f (x) which satisfy f (x) = f (c) + f ′ (c) (x − a) + g c (x) (x − c) | (x − c) | α , where x , c ∈ I and g c (x) is a continuous function for each c. We first extend the trapezoidal method on the set D α (I) and rewrite the integrand of the Caputo fractional integral as a product of two differentiable functions. In this approach, the non-smooth function and the singular kernel could have the same impact. The trapezoidal method using the Riemann–Stieltjes integral (TRSI) depends on the regularity of the two functions in the integrand. Numerical simulations demonstrated that the order of accuracy cannot be increased as the number of zones increases using the uniform discretization. However, for a fixed coarsest grid discretization, a refinable mesh approach was employed; the corresponding results show that the order of accuracy is k α , where k is a refinable scale. Meanwhile, the application of the product of two differentiable functions can also be applied to some Riemann–Liouville fractional differential equations. Finally, the stable numerical scheme is shown. [ABSTRACT FROM AUTHOR]

In this research work, time-delay adaptive synchronization and adaptive anti-synchronization of chaotic fractional order systems are analyzed via the Caputo fractional derivative, and the prob-lem of synchronization and anti-synchronization of chaotic systems of variable fractional order is solved by using the fractional order PID control law, the adaptive laws of variable-order frac-tional calculus, and a control law deduced from Lyapunov's theory extended to systems of time-delay variable-order fractional calculus. In this research work, two important problems are solved in the control area: The first problem is described in which deals with syn-chro-nization of chaotic systems of adaptive fractional order with time delay, this problem is solved by using the fractional order PID control law and adaptative laws. The second problem is de-scribed in which deals with anti-synchronization of chaotic systems of adaptive frac-tional order with time delay, and this problem is solved by using the fractional order PID con-trol law and adaptative laws. [ABSTRACT FROM AUTHOR]

Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to these initial value problems possess discontinuities at the origin. We also show how these initial value problems can be re-formulated to provide solutions that are continuous at the origin but this imposes further constraints on the system. Consideration of the intrinsic discontinuities, or constraints, in these initial value problems is important if they are to be employed in mathematical modelling applications. [ABSTRACT FROM AUTHOR]

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann-Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann-Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann-Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work. [ABSTRACT FROM AUTHOR]

We consider general linear multi-term Caputo fractional integro-differential equations with weakly singular kernels subject to local or non-local boundary conditions. Using an integral equation reformulation of the proposed problem, we first study the existence, uniqueness and regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the problem is discussed. Optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider a problem of continuity fractional-order for pseudo-parabolic equations with the fractional derivative of Caputo. Here, we investigate the stability of the problem with respect to derivative parameters and initial data. We also show that uω' → uω in an appropriate sense as ω' → ω where ω is the fractional order. Moreover, to test the continuity fractional-order, we present several numerical examples to illustrate this property. [ABSTRACT FROM AUTHOR]

In the present note, a new modification of the Adomian decomposition method is developed for the solution of fractional-order diffusion-wave equations with initial and boundary value Problems. The derivatives are described in the Caputo sense. The generalized formulation of the present technique is discussed to provide an easy way of understanding. In this context, some numerical examples of fractional-order diffusion-wave equations are solved by the suggested technique. It is investigated that the solution of fractional-order diffusion-wave equations can easily be handled by using the present technique. Moreover, a graphical representation was made for the solution of three illustrative examples. The solution-graphs are presented for integer and fractional order problems. It was found that the derived and exact results are in good agreement of integer-order problems. The convergence of fractional-order solution is the focus point of the present research work. The discussed technique is considered to be the best tool for the solution of fractional-order initial-boundary value problems in science and engineering. [ABSTRACT FROM AUTHOR]

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions. [ABSTRACT FROM AUTHOR]