Solving conformable Gegenbauer differential equation and exploring its generating function

In this manuscript, we address the resolution of conformable Gegenbauer differential equations. We demonstrate that our solution aligns precisely with the results obtained through the power series approach. Furthermore, we delve into the investigation and validation of various properties and recursive relationships associated with Gegenbauer functions. Additionally, we introduce and substantiate the conformable Rodriguez's formula and generating functio

A new non-conformable derivative based on Tsallis’s q- exponential function

Neste artigo, uma nova derivada do tipo local é proposta e algumas propriedades básicas são estudadas. Esta nova derivada satisfaz algumas propriedades do cálculo de ordem inteira, por exemplo linearidade, regra do produto, regra do quociente e a regra da cadeia. Devido à função exponencial generalizada de Tsallis, podemos estender alguns dos resultados clássicos, a saber: teorema de Rolle, teorema do valor médio. Apresentamos a correspondente Q-integral a partir da qual surgem novos resultados. Especificamente, generalizamos a propriedade de inversão do teorema fundamental do cálculo e provamos um teorema associado à integração clássica por partes. Finalmente, apresentamos uma aplicação envolvendo equações diferenciais lineares por meio da Q-derivada.In this paper, a new derivative of local type is proposed and some basic properties are studied. This new derivative satisfies some properties of integer-order calculus, e.g. linearity, product rule, quotient rule and the chain rule. Because Tsallis' generalized exponential function, we can extend some of the classical results, namely: Rolle's theorem, the mean-value theorem. We present the corresponding Q-integral from which new results emerge. Specifically, we generalize the inversion property of the fundamental theorem of calculus and prove a theorem associated with the classical integration by parts. Finally, we present an application involving linear differential equations by means of Q derivative

Time-fractional Caputo derivative versus other integro-differential operators in generalized Fokker-Planck and generalized Langevin equations

Fractional diffusion and Fokker-Planck equations are widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integro-differential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We here study the dynamics of the associated generalized Fokker-Planck equations from the perspective of the moments, the time averaged mean squared displacements, and the autocovariance functions. We also study generalized Langevin equations based on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integro-differential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integro-differential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equations for the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equations are unveiled to share similar properties with scaled and fractional Brownian motion, respectively.Comment: 26 pages, 7 figures, RevTe

Solving the Hotdog Problem by Using the Joint Zero-order Finite Hankel - Elzaki Transform

يختص هذا البحث بدمج تحويلين مختلفين معاً لتقديم تحويل مشترك جديد FHET وتحويله العكسي IFHET، كما أنَّه تمَّ إيجاد أهم خصائص FHET وإثباتها، والتي تسمى خاصية تحويل هانكل المنتهي- الزاكي لمؤثر بيسل التفاضلي، تمت مناقشة هذه الخاصة لأجل شرطين حديين مختلفين هما ديرخليه وروبين. حيث تظهر أهمية هذه الخاصة من خلال حل المعادلات التفاضلية الجزئية ذات التماثل المحوري والانتقال إلى معادلة جبرية بشكل مباشر. أيضاً تمَّ تطبيق طريقة تحويل هانكل المنتهي-الزاكي المشتركة في حل مسألة رياضية فيزيائية وهي مسألة هوت دوغ (النقانق). تمَّ مناقشة الحالة المستقرة التي لا تعتمد على الزمن لكل حل عام حصلنا عليه أي في الحالتين الغليان والتبريد. تمّ رسم الأشكال من الشكل 4 إلى الشكل 9 رسماً يدوياً على برنامج بوربوينت وذلك لتوضيح فكرة ارتفاع وانخفاض الحرارة على المجال الزمني المعطى في المسألة. تؤكد النتائج التي حصلنا عليها أن تقنية التحويل المقترحة فعالة ودقيقة وسريعة في حل المعادلات التفاضلية الجزئية المتماثلة المحور.This paper is concerned with combining two different transforms to present a new joint transform FHET and its inverse transform IFHET. Also, the most important property of FHET was concluded and proved, which is called the finite Hankel – Elzaki transforms of the Bessel differential operator property, this property was discussed for two different boundary conditions, Dirichlet and Robin. Where the importance of this property is shown by solving axisymmetric partial differential equations and transitioning to an algebraic equation directly. Also, the joint Finite Hankel-Elzaki transform method was applied in solving a mathematical-physical problem, which is the Hotdog Problem. A steady state which does not depend on time was discussed for each obtained general solution, i.e. in the boiling and cooling states. To clarify the idea of temperature rise and fall over the time domain given in the problem, some figures were drawn manually using Microsoft PowerPoint. The obtained results confirm that the proposed transform technique is efficient, accurate, and fast in solving axisymmetric partial differential equations

Stability of Finite Difference Schemes to Pseudo-Hyperbolic Telegraph Equation

Hyperbolic partial differential equations are frequently referenced in modeling real-world problems in mathematics and engineering. Therefore, in this study, an initial-boundary value issue is proposed for the pseudo-hyperbolic telegraph equation. By operator method, converting the PDE to an ODE provides an exact answer to this problem. After that, the finite difference method is applied to construct first-order finite difference schemes to calculate approximate numerical solutions. The stability estimations of finite difference schemes are shown, as well as some numerical tests to check the correctness in comparison to the precise solution. The numerical solution is subjected to error analysis. As a result of the error analysis, the maximum norm errors tend to decrease as we increase the grid points. It can be drawn that the established scheme is accurate and effectiv

NOVEL METHODS FOR SOLVING THE CONFORMABLE WAVE EQUATION

In this paper, a two-dimensional conformable fractional wave equation describing a circular membrane undergoing axisymmetric vibrations is formulated. It was found that the analytical solutions of the fractional wave equation using the conformable fractional formulation can be easily and efficiently obtained using separation of variables and double Laplace transform methods. These solutions are compared with the approximate solution obtained using the differential transform method for certain cases

A class of nonlocal impulsive differential equations with conformable fractional derivative

In this paper, we deal with the Duhamel formula, existence, uniqueness, and stability of mild solutions of a class of nonlocal impulsive differential equations with conformable fractional derivative. The main results are based on the semigroup theory combined with some fixed point theorems. We also give an example to illustrate the applicability of our abstract results

Conformable Laplace Transform of Fractional Differential Equations

In this paper, we use the conformable fractional derivative to discuss some fractional linear differential equations with constant coefficients. By applying some similar arguments to the theory of ordinary differential equations, we establish a sufficient condition to guarantee the reliability of solving constant coefficient fractional differential equations by the conformable Laplace transform method. Finally, the analytical solution for a class of fractional models associated with the logistic model, the von Foerster model and the Bertalanffy model is presented graphically for various fractional orders. The solution of the corresponding classical model is recovered as a particular case