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In this work, we study the Chafee-Infante model with conformable fractional derivative. This model describes the energy balance between equator and pole of solar system, which transmit energy via heat diffusion. To explore the multi soliton solutions and their interaction, we implemented the new modified simple equation (NMSE) scheme. Under some conditions, the obtained solutions are trigonometric, hyperbolic, exponential and their combine form. Only the proposed technique can be provided the solution in terms of trigonometric and hyperbolic form together directly. The periodic, solitary wave and novel interaction of such solitary and sinusoidal solutions has also been established and discussed analytically. For the special values of the existing free parameter, some novel waveforms are existed for the proposed model including, periodic solution, double periodic wave solution, multi-kink solution. The behavior of the obtained solutions is presented in 3-D plot, density plot and counter plot with the help of computational software Maple 18., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Authors.)

The motivation of this investigation is to develop a single-layer Chebyshev Neural Network (ChNN) model to handle singular fractional (arbitrary)-order Lane-Emden type equations. These equations are well-known application problems of astrophysics and quantum mechanics. Fractional Lane-Emden equations are singular so it is very difficult to solve analytically. Thus, an efficient method is required to handle the above equations. Here, our main aim is to use a single-layer ChNN model for solving fractional Lane-Emden equations. ChNN model is one kind of Functional Link Neural Network (FLNN) in which the hidden layer is replaced by a functional expansion block of the input pattern using orthogonalshifted Chebyshev polynomials (SChP). Thus, the network parameters of ChNN are less than the Multi-Layer Artificial Neural Network (MLANN). We have considered factional-order singular nonlinear problems of astrophysics to show the computational effort of the proposed method. Back Propagation algorithm of the unsupervised version has been considered for minimizing the error function and updating the weights of the ChNN model. Computed results are displayed in terms of tables and graphs.

We establish some new noninstantaneous impulsive inequalities using the conformable fractional calculus., Competing Interests: The authors declare that they have no competing interests.

We study the conformable fractional (CF) Dirac system with separated boundary conditions on an arbitrary time scale [Formula: see text]. Then we extend some basic spectral properties of the classical Dirac system to the CF case. Eventually, some asymptotic estimates for the eigenfunction of the CF Dirac eigenvalue problem are obtained on [Formula: see text]. So, we provide a constructive procedure for the solution of this problem. These results are important steps to consolidate the link between fractional calculus and time scale calculus in spectral theory.