The Generalized Conformable Derivative for 4$alpha$-Order Sturm-Liouville Problems

A generalized differential operator on the real line is defined by means of a limiting process. When an additional fractional parameter is introduced, this process leads to a locally defined fractional derivative. The study of such generalized derivatives includes, as a special case, basic results involving the classical derivative and current research involving fractional differential operators. All our operators satisfy properties such as the sum, product/quotient rules, and the chain rule. We study a Sturm–Liouville eigenvalue problem with generalized derivatives and show that the general case is actually a consequence of standard Sturm–Liouville Theory. As an application of the developments herein we find the general solution of a generalized harmonic oscillator. We also consider the classical problem of a planar motion under a central force and show that the general solution of this problem is still generically an ellipse, and that this result is true independently of the choice of the generalized derivatives being used modulo a time shift. The generic nature of phase plane orbits modulo a time shift is extended to the classical gravitational n-body problem of Newton to show that the global nature of these orbits is independent of the choice of the generalized derivatives being used in defining the force law. Finally, restricting the generalized derivatives to a special class of fractional derivatives, we consider the question of motion under gravity with and without resistance and arrive at a new notion of time that depends on the fractional parameter. The results herein are meant to clarify and extend many known results in the literature and intended to show the limitations and use of generalized derivatives and corresponding fractional derivatives.

In this paper, we study the inverse problem for Sturm-Liouville problem with conformable fractional differential operators of order $\alpha$, $0.5 < \alpha\leq 1$ and finite number of interior discontinuous conditions. For this aim first, the asymptotic formulas for solutions, eigenvalues and eigenfunctions of the problem are calculated. Then some uniqueness theorems for proposed inverse eigenvalue problem are proved. Finally, the Hald's theorem for conformable Sturm-Liouville problem is developed.

Derivative-orthogonal Riesz wavelets in Sobolev spaces with applications to differential equations

A chain rule for power product is studied with fractional differential operators in the framework of Sobolev spaces. The fractional differential operators are defined by the Fourier multipliers. The chain rule is considered newly in the case where the order of differential operators is between one and two. The study is based on the analogy of the classical chain rule or Leibniz rule.

This paper addresses the numerical solutions of fractional differential equations (FDEs) using the Generalized Kudryashov Method (GKM) in the context of the conformable fractional derivative. Fractional calculus, particularly the conformable derivative, provides a versatile framework for modeling systems exhibiting memory and hereditary properties commonly found in complex physical phenomena. Traditional integer-order derivatives lack the capability to accurately represent such dynamics, which fractional derivatives effectively handle. The conformable derivative, a recent addition to fractional calculus, retains many advantageous properties of integer-order differentiation, such as the chain rule, while extending to non-integer orders. The Generalized Kudryashov Method, initially developed for solving nonlinear ordinary differential equations, is adapted here to address nonlinear FDEs involving conformable derivatives. By employing a traveling wave transformation, the study converts fractional partial differential equations into ordinary differential equations, facilitating the application of GKM. Through this approach, the study derives numerical solutions, demonstrating the method’s ability to capture complex dynamics in nonlinear fractional systems. The results indicate that GKM, in conjunction with the conformable derivative, offers a robust tool for accurately approximating solutions of FDEs, with potential applications across fields such as fluid mechanics, quantum mechanics, and anomalous diffusion.

Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative

The concept of a conformable derivative on time scales is a relatively new development in the field of fractional calculus. Traditional fractional calculus deals with derivatives and integrals of non-integer order on continuous time domains. However, time scale calculus extends these concepts to more general time domains that include both continuous and discrete points. The conformable derivative on time scales has several properties that make it advantageous in certain applications. For example, it satisfies a chain rule and has a simple relationship with the conformable integral, which facilitates the development of differential equations involving fractional order dynamics. It also allows for the analysis of systems with both continuous and discrete data points, making it suitable for modeling and control applications in various fields, including physics, engineering, and finance. In this study, the Sturm-Liouville problem and its properties are examined on an arbitrary time scale using the proportional derivative, a more general form of the fractional derivative. Important spectral properties such as self-adjointness, Green formula, Lagrange identity, Abel formula, and orthogonality of eigenfunctions for this problem are expressed in proportional derivatives on an arbitrary time scale.

Simplest equation method for some time-fractional partial differential equations with conformable derivative

In the paper, we study calculus rules of second-order composed contingent derivatives. More precisely, chain rule and sum rule are established and their applications to some particular mathematical models are obtained. Then sensitivity analysis in set-valued optimization using second-order composed contingent derivatives are proposed. Our results are new and many examples are given to illustrate them.

In this paper, we will give the derivation of an inquiry calculus, or, equivalently, a Bayesian information theory. From simple ordering follow lattices, or, equivalently, algebras. Lattices admit a quantification, or, equivalently, algebras may be extended to calculi. The general rules of quantification are the sum and chain rules. Probability theory follows from a quantification on the specific lattice of statements that has an upper context. Inquiry calculus follows from a quantification on the specific lattice of questions that has a lower context. There will be given here a relevance measure and a product rule for relevances, which, taken together with the sum rule of relevances, will allow us to perform inquiry analyses in an algorithmic manner.

A numerical method for fractional Sturm–Liouville problems involving the Cauchy–Euler operators

In this article, we study Sturm-Liouville Equations (SLEs) in the frame of fractional operators with exponential kernels. We formulate some Fractional Sturm-Liouville Problems (FSLPs) with the differential part containing the left and right sided derivatives with exponential kernels. We investigate the self-adjointness, eigenvalue and eigenfunction properties of the corresponding Fractional Sturm-Liouville Operators (FSLOs) by using fractional integration by parts formulas. The nabla discrete version of our results are also established. Finally, an example is analyzed to illustrate the method of solution.

In this paper, we proved transformation operator for fractional Sturm-Liouville operator, using conformable derivative approach, which is different from classical Sturm-Liouville operator. Especially, we obtained a Hyperbolic partial differential equation and some suitable conditions for nucleus function K(x, t). Finally, we obtained a Fredholm integral equation. The proof is validated by taking ? = 1 which returns the original problem.

A survey of results on Lyapunov-type inequalities for fractional differential equations associated with a variety of boundary conditions is presented. This includes Dirichlet, mixed, Robin, fractional, Sturm–Liouville, integral, nonlocal, multi-point, anti-periodic, conjugate, right-focal, and impulsive conditions. Furthermore, our study includes Riemann–Liouville, Caputo, Hadamard, Prabhakar, Hilfer, and conformable type fractional derivatives. Results for boundary value problems involving fractional p-Laplacian, fractional operators with nonsingular Mittag–Leffler kernels, q-difference, discrete, and impulsive equations are also taken into account.

This paper presents a new technique: a conformable derivative for the inverse problem of a Sturm-Liouville problem with restrained constant delay. Solutions to the Sturm-Liouville problem often involve eigenfunctions and eigenvalues, which have important applications in physics, engineering, and other fields. The presence of a constant delay introduces unique challenges in formulating and solving this problem. In this case, we derived the asymptotic formulas for the eigenvalues with their corresponding eigenfunctions and demonstrated the existence of the solution. Additionally, we identified the nodal points used to generate the problem’s potential function. Finally, we applied the Lipschitz stability approach and demonstrated the stability of the solution to the problem.