The integral-order derivative is not suitable where infinite variances are expected, and the fractional derivative manages to consider effects with more precision; therefore, we considered timefractional Emden–Fowler-type equations and solved them using the rational homotopy perturbation method (RHPM). The RHPM method is based on two power series in rational form. The existence and uniqueness of the equation are proved using the Banach fixed-point theorem. Furthermore, we approximate the term h(z) with a polynomial of a suitable degree and then solve the system using the proposed method and obtain an approximate symmetric solution. Two numerical examples are investigated using this proposed approach. The effectiveness of the proposed approach is checked by representing the graphs of exact and approximate solutions. The table of absolute error is also presented to understand the method′s accuracy.

In this paper, a deep artificial neural network technique is proposed to solve the coupled system of Emden-Fowler equations. A vectorized form of algorithm is developed. Implementation and simulation of this technique is performed using Python code. This technique is implemented in various numerical examples, and simulations are conducted. We have shown graphically how accurately this method works. We have shown the comparison of numerical solution and exact solution using error tables. We have also conducted a comparative analysis of our solution with alternative methods, including the Bernstein collocation method and the Homotopy analysis method. The comparative results are presented in error tables. The efficiency and accuracy of this method are demonstrated by these graphs and tables.

This paper aims to solve general fractional Lane-Emden-Fowler differential equations using the Haar wavelet collocation method. This method transforms the fractional differential equation into a nonlinear system of equations, which is further solved for Haar coefficients using Newton’s method. We have constructed the higher-order Lane-Emden-Fowler equations. We have also discussed the convergence rate and stability analysis of our technique. We have explained the applications and numerically simulated the examples graphically and in tabular format to elaborate on the accuracy and efficiency of this approach.

The present study introduces the Haar wavelet method, which utilizes collocation points to approximate solutions to the Emden-Fowler Pantograph delay differential equations (PDDEs) of general order. This semi-analytic method requires the transformation of the original differential equation into a system of nonlinear differential equations, which is then solved to determine the Haar coefficients. The method’s application to fourth-, fifth-, and sixth-order PDDEs is discussed, along with an examination of convergence that involves the determination of an upper bound and the formulation of the rate of convergence for the method. Numerical simulations and error tables are presented to demonstrate the effectiveness and precision of this approach. The error tables clearly illustrate that the method’s accuracy improves progressively with increasing resolution.

In this study, we introduce an algorithm that utilizes the Haar wavelet collocation method to solve the time-dependent Emden–Fowler equation. This proposed method effectively addresses both linear and nonlinear partial differential equations. It is a numerical technique where the differential equation is discretized using Haar basis functions. A difference scheme is also applied to approximate the time derivative. By leveraging Haar functions and the difference scheme, we form a system of equations, which is then solved for Haar coefficients using MATLAB software. The effectiveness of this technique is demonstrated through various examples. Numerical simulations are performed, and the results are presented in graphical and tabular formats. We also provide a convergence analysis and an error analysis for this method. Furthermore, approximate solutions are compared with those obtained from other methods to highlight the accuracy, efficiency, and computational convenience of this technique.