This paper introduces an analytical framework for solving the coupled fractional Whitham–Broer–Kaup (WBK) equations. This framework integrates the Sumudu Transform (ST) with the Decomposition Method (DM), employing fractional derivatives defined in the Caputo (C), Atangana–Baleanu Caputo (ABC), and Caputo–Fabrizio (CF) senses. The Sumudu Transform serves as a potent analytical instrument, streamlining the solution process and upholding both initial and boundary conditions. Our methodology’s precision and convergence are validated through the derivation of exact and approximate solutions for WBK equations. A thorough comparison with existing methods confirms the simplicity and physical validity of the SDM framework. This research advances fractional modeling and bridges a gap in Caputo (C), Atangana–Baleanu Caputo (ABC), and Caputo–Fabrizio (CF) fractional calculus by incorporating integral transform methods. It offers a new perspective for tackling complex nonlinear fractional partial differential equations (PDEs). The evidence presented in the tables and figures collectively substantiates that the proposed method approaches are highly precise, robust, and adaptable across diverse fractional orders, rendering them suitable for investigating intricate waves and nonlinear dispersive systems.

In this paper, we introduce a novel modification of the fractional conformable triple Sumudu transformation and subsequently establish fundamental theorems that pave the way for various practical applications. Our theoretical discoveries contribute significantly to the advancement of this field by creating innovative avenues for tackling nonlinear partial fractional differential equations, particularly within the framework of the Adomian Sumudu decomposition method.

This study presents a numerical solution of the Bratu differential equations (BDE) using the Sumudu transform series decomposition technique (STSDT). The process combines the Adomian polynomials (AP), series expansion (SE), and Sumudu transform (ST), and it ultimately converges perfectly to the exact solution. Examining four test problems demonstrates that the strategy converges more effectively than the literature-based approach. Calculations were performed using Maple 2022 software.

The employing of mathematical models to comprehend complicated processes is widespread. The mathematical model often takes the form of an ordinary differential equation system. The Susceptible–Infected–Recovered (SIR) epidemic model of childhood diseases is mathematically analyzed and reported in this research paper using the Caputo fractional-order derivative operator. The series/approximate solutions of the aforesaid model are explained with the help of the Sumudu Transform Homotopy Perturbation Method (STHPM). To verify our findings, we presented three graphs for the susceptible class [Formula: see text], infected [Formula: see text] and recovered [Formula: see text] and noted that for smaller values of [Formula: see text], the susceptible class declined more slowly indicating a stronger memory effect and slower spread of the epidemic. For the infected class, smaller values of [Formula: see text] increase a decay in infection, while higher values of [Formula: see text] make the system resemble the classical integer-order model. This suggests that the proposed fractional-order model is more suitable for childhood diseases due to the memory effects, flexibility of predictive modeling of the epidemic and biological consistency of the initial conditions. The Caputo fractional derivative operator and the projected technique (STHPM) are expected to open up new avenues for manipulating and analyzing a number of epidemiological models in future research.

In this paper, a linear second-order non-homogeneous fuzzy partial differential equation (FPDE) is constructed, and the Fuzzy Sumudu ‎Transform (FST) method is applied to solve FPDEs within the context of generalized Hukuhara(gH) differentiability technique. The use ‎of FST, a potent integral transform renowned for its scale-invariant and unit-preserving characteristics, to the fuzzy setting is expanded. ‎FPDEs are solved analytically by transforming them into more straightforward algebraic differential equations in the transform domain, ‎utilizing recent advances in the gH-differentiability technique. Initially, the basic characteristics of linear second-order non-homogeneous FPDEs are presented. To highlight the capabilities, a numerical example is provided.

This article discusses the solution of the Burgers equation, which is a nonlinear partial differential equation, particularly for the 3D Burgers equation. This equation will be solved using a combination of the Homotopy Perturbation Method (HPM) and the Sumudu Transform (ST), known as HPM-ST. HPM-ST an alternative method to those found in the existing literature. This method is effective and easy to determine the analytic solution of nonlinear equations. To implement HPM-ST, the Sumudu transform and inverse Sumudu transform are applied first, so a nonlinear differential equation is obtained that does not depend on the variable t. Then, HPM is applied to this equation to derive an infinite series, which can be approximated using a Maclaurin series. The analytical solution of the 3D Burgers equation obtained by HPM-ST is equivalent to the exact solution. To provide an overview of the solution of the 3D Burgers equation, a visualization of the obtained solution is also presented using MATLAB.

This study presents a novel approach to solving fractional-order kinetic equations (FKEs) by employing Hyper-Bessel functions in conjunction with the Sumudu transform. The proposed method leverages the Sumudu transform to obtain analytical solutions of FKEs, wherein the function f(t) is expressed in terms of a Hyper-Bessel function. The resulting solutions are general in nature and can be applied to a wide range of existing as well as newly formulated fractional kinetic equations, with potential applications in environmental science and related fields.Special functions included in fractional kinetic equations have been shown to be helpful in the explanation and resolution of numerous important mathematical and mathematical physics issues. Because arbitrary-order kinetic equations are so important, the goal of this study is to use the Sumudu Transform technique to solve a new fractional-order kinetic equation involving the Hyper Bessel function with their fractional derivatives. MATLAB-generated graphical representations used in our analysis to demonstrate the behavior of these solutions under various parametric values. The outcomes of the study are highly flexible and may lead to both confirmed and maybe undiscovered research findings in this field.

The aim of this study is to demonstrate the properties of the non-Newtonian Sumudu transform, which are required to investigate the solutions of non-Newtonian differential equations with variable coefficients by using the non-Newtonion Sumudu transform. The use of the obtained results in determining the solutions of these equations is supported by numerical examples.

This work reveals an advanced numerical scheme for obtaining approximate solutions to nonlinear fractional Kuramoto–Sivashinsky (K-S) equations involving Caputo derivatives. We introduce the Sumudu transform (ST), which converts the fractional derivatives into their classical counterparts to produce a nonlinear recurrence equation. By using the homotopy perturbation method (HPM), we construct a homotopy with an embedding parameter to solve this recurrence relation. Our proposed technique is known as the Sumudu homotopy transform method (SHTM), which delivers results after fewer iterations and achieves precise outcomes with minimal computational effort. The proposed technique effectively eliminates the necessity for complex discretization or linearization, making it highly suitable for nonlinear problems. We showcase two numerical cases, along with two- and three-dimensional visualizations, to validate the accuracy and effectiveness of this technique. It also produces rapidly converging series solutions that closely align with the precise results.

In today’s world, information security is the most important component of our lives. Cryptography is among the best techniques designed to safeguard data security and message transmission. This paper presents a new method using successive combinations of the Sumudu transform of three functions for encoding and the inverse Sumudu transform for decoding. Starting with standard results on Sumudu transforms, we present our encryption method and obtain it as new theorems. Further, we generalized our method for the more secure algorithm. Finally, we used Python code to implement this strategy with simulation, our conclusions are based on the statistical analysis.

Abstract In this study, we propose an efficient iterative technique for investigating the analytical approximate solutions of the Caputo derivative-based fractional diffusion-wave equations. The proposed technique involves the Sumudu transform (ST) and the Daftardar-Gejji and Jafari method (DGJM). Further, we give various illustrations of linear and nonlinear diffusion-wave equations and demonstrate the solutions figuratively. The proposed technique is free from complex calculations and works without any discretization.

The concept of time-fractional Emden–Fowler (EF) model is used to describe the fluid flow in porous media, where the flow displays fractional-order behavior due to intricate microstructures. This study aims to analyze the fractional analysis of a linear and nonlinear time-fractional EF model, in which the fractional derivatives are taken in the Caputo form. We establish the Sumudu perturbation transform method (SPTM) by combining the Sumudu transform (ST) with the homotopy perturbation method (HPM). The ST discretizes the fractional-order model to a Sumudu space where the utilization of HPM handles the nonlinear parameters in the recurrence relation very effectively. We provide two numerical tests of the time-fractional EF model to check the legitimacy and reliability of the suggested scheme. The derived outcomes by SPTM enhance the excellence over a longer period by reducing the parameters of the truncated series. Some graphical representations at different fractional orders are provided to validate our proposed scheme. This approach offers a clear and concise procedure with symbolic computation to derive the series results which converge very promptly.

In this study, we employ fuzzy Sumudu transform to find the solution for system of linear fuzzy differential equations where the system possesses fuzzy constant coefficients instead of crisp.For this purpose, fuzzy Sumudu transform has been revisited and a brief comparison with fuzzy Laplace transform is provided alongside, particularly on the scale preserving property.For the sake of comparison, we introduce to the literature a time scaling theorem for fuzzy Laplace transform.Next, the system with fuzzy constant coefficients is interpreted under the strongly generalized differentiability.From here, new procedures for solving the systems are proposed.A numerical example is then carried out for solving a system adapted from fuzzy radioactive decay model.Conclusion is drawn in the last section and some potential research directions are given.

Solving Nonlinear Volterra-Fredholm Integro-Differential Equations of the Second Kind by Combining the Sumudu Transform and the Adomian Decomposition Method

The authors of the article discussed the duality relations of fractional Laplace transforms with kamal, Laplace-Carson, Aboodh, Sumudu, Elazaki, Mohand, Sawi, and natural transforms of fractional orders in this article. Tabular Presentation of Laplace Transforms, Laplace-Carson Transforms, Aboodh Transforms, Sumudu Transforms, Natural Transforms, Elazaki Transforms, Mohand and Sawi Transforms of Fractional Orders of Some Basic Mathematical Functions.

Time Fractional Black–Scholes Model and Its Solution Through Sumudu Transform Iterative Method

Fuzzy differential equations (FDEs) are used to represent dynamical systems under uncertain environments. Finding solutions for fuzzy differential equations (FDEs) is highly challenging. This work employs the neutrosophic version of the Sumudu transform method to determine the solution to fuzzy differential equations (FDEs) that incorporate Neutrosophic Numbers (NNs). By utilising a novel fuzzy arithmetic operations on the parametric representations of NNs, significant theorems are established to demonstrate the characteristics of Neutrosophic Sumudu Transform (NST). The proposed NST approach is efficient in approximating the solutions of FDEs without converting them into their crisp equivalent forms. An illustrative numerical example is provided to demonstrate the efficacy of the proposed methodology.

This research introduces the nonconformable fractional Sumudu transform (NCFST) methodology. We used the above strategy to solve fractional differential equations (FDEs) via nonconformable fractional derivatives (NCFDs). We examined and demonstrated its fundamental qualities and benefits. We examined several cases and performed a comparison analysis, delivering precise data via graphs and tables. The findings indicated that the novel approach was practical, user‐friendly, and capable of accurately solving FDEs, including those with NCFDs.

In this article, we present the nonconformable fractional derivative of the double Sumudu transformation. In this study, we investigate the main features and benefits of this new technique and then apply it to solve several fractional nonconformable partial differential equations. To test the validity of the latest suggested method and its efficiency, we consider three different examples and perform a comparative study with the exact results. We also use numerical tests to ensure the proposed method is correct and useful. The graphs and tables we choose to indicate how the method works in different fractional situations match the exact results. The results indicate that the new method is effective and simple to use for finding suitable solutions to various types of fractional partial differential equations, including those with nonconformable fractional derivatives.

By combining the new iteration method (NIM) and Sumudu transform (ST) methods, together known as the new iteration ST method (NISTM), a semi-analytical or series solution for various fractional differential equations, in which new iteration method (NIM) is used to decompose the nonlinear operator prior to performing the ST, is produced in this script. Using the Caputo sense to account for the fractional derivative, this method computes series solutions for a variety of nonlinear fractional differential equation instances. While in many convergence issues a solution is achieved for just a small number of series terms, our series solution merges with the exact solution differential equation of fractional order in many example problems.