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In this study, we introduce a novel iterative method combined with the Elzaki transformation to address a system of partial differential equations involving the Caputo derivative. The Elzaki transformation, known for its effectiveness in solving differential equations, is incorporated into the proposed iterative approach to enhance its efficiency. The system of partial differential equations under consideration is characterized by the presence of Caputo derivatives, which capture fractional order dynamics. The developed method aims to provide accurate and efficient solutions to this complex mathematical system, contributing to the broader understanding of fractional calculus applications in the context of partial differential equations. Through numerical experiments and comparisons, we demonstrate the efficacy of the proposed Elzaki-transform-based iterative method in handling the intricate dynamics inherent in the given system. The study not only showcases the versatility of the Elzaki transformation but also highlights the potential of the developed iterative technique for addressing similar problems in various scientific and engineering domains.

This article demonstrates how the new Double Laplace–Sumudu transform (DLST) is successfully implemented in combination with the iterative method to obtain the exact solutions of nonlinear partial differential equations (NLPDEs) by considering specified conditions. The solutions of nonlinear terms of these equations were determined by using the successive iterative procedure. The proposed technique has the advantage of generating exact solutions, and it is easy to apply analytically on the given problems. In addition, the theorems handling the mode properties of the DLST have been proved. To prove the usability and effectiveness of this method, examples have been given. The results show that the presented method holds promise for solving other types of NLPDEs.

In this paper, the new iterative transform method and the homotopy perturbation transform method was used to solve fractional-order Equal-Width equations with the help of Caputo-Fabrizio. This method combines the Laplace transform with the new iterative transform method and the homotopy perturbation method. The approximate results are calculated in the series form with easily computable components. The fractional Equal-Width equations play an essential role in describe hydromagnetic waves in cold plasma. Our object is to study the nonlinear behaviour of the plasma system and highlight the critical points. The techniques are very reliable, effective, and efficient, which can solve a wide range of problems arising in engineering and sciences.

In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in series form to analyze the analytical results of fractional-order coupled Korteweg-de Vries equations. To understand the analytical procedure of Iteration transform method, some numerical problems are presented for the analytical result of fractional-order coupled Korteweg-de Vries equations. It is also demonstrated that the current technique’s solutions are in good agreement with the exact results. The numerical solutions show that only a few terms are sufficient for obtaining an approximate result, which is efficient, accurate, and reliable.

In science and engineering, nonlinear time‐fractional partial differential equations (NTFPDEs) are thought to be a useful tool for describing several natural and physical processes. It is tough to come up with analytical answers for these issues. Finding answers to NTFPDEs is therefore a crucial component of scientific study. To solve nonlinear time‐fractional hyperbolic partial differential equations (NTFHPDEs), we provide a novel fractional Shehu transform iterative method (FSTIM) in this work. In this case, the Caputo derivative is used to get the fractional derivative. The method combines two powerful numerical approaches called the fractional Shehu transform and the new iterative method (NIM), also known as the Daftardar‐Gejji and Jafari method. Through the recommended scheme, the linear portion of the problem is resolved by employing the Shehu transform method, while the noise terms from the nonlinear portion of the problem disappear over a successive iteration process of the NIM. The solution of FSTIM is then denoted in a series form, which is convergent to the precise answer of the assumed problem. Using principles from Banach’s spaces, the stability and convergence analysis of the suggested approach are addressed. To confirm the effectiveness and accuracy of the method, three illustrations from NTFHPDEs are presented. The obtained results are compared with the exact solutions and the other numerical results existing in the literature in terms of L ∞ and L 2 absolute errors. The findings showed that the proposed method outperforms the other numerical techniques in the literature and gives accurate results with a few terms. Therefore, the recommended approach is effective and straightforward and can be applied to other complex nonlinear physical differential equations with fractional order.

The study of fractional-order partial differential equations has gained significant attention due to its ability to model complex physical systems with memory effects and hereditary properties. Among these systems is the Burgers equation, which serves as a fundamental model for describing nonlinear waves, turbulence, and the behavior of viscous fluids. Recent advancements in fractional calculus have led to the development of generalized fractional operators, such as the Caputo-Hadamard and ϕ -Caputo fractional derivatives. These operators offer greater flexibility and precision in capturing temporal and spatial nonlocal effects. This paper provides a comprehensive review of analytical and semi-analytical methods for solving these systems, with a particular emphasis on the Residual Power Series Method (RPSM) and the New Iterative Method (NIM). Both methods demonstrate superior convergence rates and accuracy. By combining generalized fractional operators with iterative algorithms, new approaches can be developed to model real-world phenomena in fields like physics, engineering, and applied mathematics. This work not only summarizes recent progress but also establishes a strong foundation for future research into complex fractional-order systems.

This study analyzes the family of one of the most essential fractional differential equations due to its wide applications in physics and engineering: the multidimensional fractional linear and nonlinear diffusion equations. The Caputo fractional derivative operator is used to treat the time-fractional derivative. To complete the analysis and generate more stable and highly accurate approximations of the proposed models, three extremely effective techniques, known as the direct Tantawy technique, the new iterative transform technique (NITM), and the homotopy perturbation transform method (HPTM), which combine the Elzaki transform (ET) with the new iterative method (NIM), and the homotopy perturbation method (HPM), are employed. These reliable approaches produce more stable and highly accurate analytical approximations in series form, which converge to the exact solutions after a few iterations. As the number of terms/iterations in the problems series solution rises, it is found that the derived approximations are closely related to each problem’s exact solutions. The two- and three-dimensional graphical representations are considered to understand the mechanism and dynamics of the nonlinear phenomena described by the derived approximations. Moreover, both the absolute and residual errors for all generated approximations are estimated to demonstrate the high accuracy of all derived approximations. The obtained results are encouraging and appropriate for investigating diffusion problems. The primary benefit lies in the fact that our proposed plan does not necessitate any presumptions or limitations on variables that might affect the real problems. One of the most essential features of the proposed methods is the low computational cost and fast computations, especially for the Tantawy technique. The findings of the present study will be valuable as a tool for handling fractional partial differential equation solutions. These approaches are essential in solving the problem and moving beyond the restrictions on variables that could make modeling the problem challenging.

The human immunodeficiency virus (HIV) mainly attacks CD4+ T cells in the host. Chronic HIV infection gradually depletes the CD4+ T cell pool, compromising the host’s immunological reaction to invasive infections and ultimately leading to acquired immunodeficiency syndrome (AIDS). The goal of this study is not to provide a qualitative description of the rich dynamic characteristics of the HIV infection model of CD4+ T cells, but to produce accurate analytical solutions to the model using the modified iterative approach. In this research, a new efficient method using the new iterative method (NIM), the coupling of the standard NIM and Laplace transform, called the modified new iterative method (MNIM), has been introduced to resolve the HIV infection model as a class of system of ordinary differential equations (ODEs). A nonlinear HIV infection dynamics model is adopted as an instance to elucidate the identification process and the solution process of MNIM, only two iterations lead to ideal results. In addition, the model has also been solved using NIM and the fourth order Runge–Kutta (RK4) method. The results indicate that the solutions by MNIM match with those of RK4 method to a minimum of eight decimal places, whereas NIM solutions are not accurate enough. Numerical comparisons between the MNIM, NIM, the classical RK4 and other methods reveal that the modified technique has potential as a tool for the nonlinear systems of ODEs.

Recently, researchers have been interested in studying fractional differential equations and their solutions due to the wide range of their applications in many scientific fields. In this paper, a new approach called the Hussein–Jassim (HJ) method is presented for solving nonlinear fractional ordinary differential equations. The new method is based on a power series of fractional order. The proposed approach is employed to obtain an approximate solution for the fractional differential equations. The results of this study show that the solutions obtained from solving the fractional differential equations are highly consistent with those obtained by exact solutions.

A new method to solve non-linear fractional-order differential equations involving delay has been presented. Applications to a variety of problems demonstrate that the proposed method is more accurate and time efficient compared to existing methods. A detailed error analysis has also been given.