Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated series of the normalized shifted Jacobi polynomials under the assumption that the exact solution is an element in L2. The spectral collocation method is used as a solver to obtain the unknown expansion coefficients. The Jacobi roots are used as collocation nodes. Our solutions can easily be a generalization of the solutions of the classical LE equation, by obtaining a numerical solution based on new parameters, by fixing these parameters to the classical case, we obtain the solution of the classical equation. We provide a meticulous convergence analysis and demonstrate rapid convergence of the truncation error concerning the number of retained modes. Numerical examples show the effectiveness and applicability of the method. The primary benefits of the suggested approach are that we significantly reduce the complexity of the underlying differential equation by solving a nonlinear system of algebraic equations that can be done quickly and accurately using Newton’s method and vanishing initial guesses.

/pdf/spectral-collocation-approach-via-normalized-shifted-jacobi-1n90bjth.pdf

Spectral Collocation Approach via Normalized Shifted Jacobi Polynomials for the Nonlinear Lane-Emden Equation with Fractal-Fractional Derivative

In this study, we generated a novel functional matrix using Bernoulli wavelets. Also, we developed a novel technique called the Bernoulli wavelets collocation method to obtain reasonably accurate solutions for the HIV-infection model of CD4+ T cells. This mathematical model is in the form of a system of a nonlinear ordinary differential equation (ODE). This approach obtains the solution for this model by transforming it into a system of nonlinear algebraic equations by expanding through Bernoulli wavelets with unknown coefficients. The collocation scheme is used to calculate these unknown coefficients. The consistency and proficiency of the developed approach are demonstrated through tables and graphs. Obtained results reveal that the current approach is more accurate than other methods in the literature. All computations have been made with the help of Mathematica software. Some properties of Bernoulli wavelets are discussed in terms of theorems. 

Modified Bernoulli wavelets functional matrix approach for the HIV infection of CD4+ T cells model

Problem statement: The study offers theoretical formulations for high-viscosity particulate flows in inclined reservoirs, taking into account the presence of homogeneous spheroidal particles of various types to produce discrete two-phase suspensions. Purpose: The primary objective of this analytical and comparative study is to identify the most dependable nanoparticles among hafnium and crystal metals that are suspended in an Eyring–Powell fluid through an inclined channel while being subjected to external magnetic and gravitational forces. Solution methodology: The flow dynamics of multiphase flows are formulated utilizing the stress tensor of the base fluid. The regular perturbation method (RPM) is employed to attain a more closed-form solution. The perturbation method is frequently employed in engineering problems to obtain an approximated solution, even when demonstrating the convergence of the solution is challenging. The rough solution is also validated through a thorough parametric analysis that shows the role of relevant parameters that contribute to the multiphase flow. Results: A concise parametric study is carried out against some of the most pertinent parameters and reveals that additional particles have promising effects on the momentum of each multiphase flow, whereas Eyring–Powell multiphase suspensions lessen in momentum due to strong internal viscous forces. The velocity of fluid and particle phases diminish with Hartmann number M and Froude number Fr. The second-order material constant B and concentration of nanoparticles C boost the motion of the fluid. The velocities of the particulate phase are quicker than the fluid phase. The hafnium particle is more reliable than crystal particles. Solution benchmark: Numerical and graphical findings have also been compared with the existing literature for the limiting case and found to be fully in agreement. Applications: This study’s findings provide a wider understanding of subterranean flows, specifically within the petroleum sector, with a focus on multiphase flows. Originality: The current study represents the authors’ original work and has not been previously submitted or published elsewhere.

Tracking Multiphase Flows through Steep Reservoirs with External Constraint

Abstract We develop a precise and efficient ultraspherical wavelet method for a famous Benjamin-Bona-Mahony (BBM) mathematical model. The suggested technique uses the collocation method and ultraspherical wavelets. The proposed scheme is applied to linear and nonlinear BBM equations to inspect the proposed technique’s efficiency and accuracy. This practical approach’s effectiveness has been verified. Moreover, the method based on the ultraspherical wavelets is simple, accurate, fast, flexible, and convenient. The results are analyzed using tables and graphs and compared with other methods in the literature. As we know, Many PDEs don’t have exact solutions, and some semi-analytical methods work based on controlling parameters, but this is a controlling parameter-free technique. Also, it is pretty simple to implement and consumes less time to execute the programs. The recommended wavelet-based numerical approach is interesting, productive, and efficient. The proposed technique’s convergence analysis is presented through the theorem.

A novel approach for Benjamin-Bona-Mahony equation via ultraspherical wavelets collocation method

Numerical Comparison of Nonlinear Thermal Radiation and Chemically Reactive Bio-convection Flow of Casson-Carreau Nano-liquid with Gyro-tactic Microorganisms: Lie Group Theoretic Approach

Numerical solution of Advection–Diffusion Equation using Graph theoretic polynomial collocation method

The current work provides a theoretical investigation of permeability effects on channel walls. The entropy generation in an incompressible Casson fluid flowing through an inclined penetrable channel which is under the magnetic action is numerically investigated by transforming the governing equations into a system of ODEs using non-dimensional parameters. The Hermite wavelet method is used to solve the achieved nonlinear coupled equations. Velocity, temperature, and entropy generation profiles are analyzed graphically and in addition, graphs are employed to assess the Bejan number, Nusselt number, and skin friction coefficient for a scale of various values of physical parameters. The results are evident that for increasing the Biot number, Grashof number, and Casson parameter entropy generation increases. The entropy production is suppressed when the Eckert number is enhanced. The dual nature of entropy generation is observed due to varying magnetic values. Our findings obtained from the present method are compared with other results available in the literature and is found that it is in good correlation with the existing one. 

Entropy generation on an MHD Casson fluid flow in an inclined channel with a permeable walls through Hermite wavelet method

Fibonacci wavelets approach for the fractional Rosenau-Hyman equations

This paper generated the novel approach called the Clique polynomial method (CPM) using the Clique polynomials raised in graph theory. Nonlinear initial value problems are converted into nonlinear algebraic equations by discretization with suitable grid points in the current approach. We solved highly nonlinear initial value problems using the Adomian decomposition method (ADM) and CPM. Obtained results reveal that the present technique is better than ADM that is discussed through tables and simulations. 

Comparative study of Adomian decomposition method and Clique polynomial method

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