This study addresses a nonlinear fractional Drinfeld–Sokolov–Wilson problem in dispersive water waves, which requires appropriate numerical techniques to obtain an approximative solution. The Adomian decomposition approach, the homotopy perturbation method, and Sumudu transform are combined to tackle the problem. The Caputo manner is used to describe fractional derivative, and He’s polynomials and Adomian polynomials are employed to address nonlinearity. By following these approaches, we obtain solutions in the form of convergent series. We verify and demonstrate the effectiveness of our suggested strategies by examining the assumed model in terms of fractional order. We use plots for various fractional orders to represent the physical behavior of the suggested technique solutions, and show a numerical simulation. The results demonstrate that the suggested algorithms are systematic, simple to use, effective, and accurate in analyzing the behavior of coupled nonlinear differential equations of fractional order in related scientific and engineering fields.

In this paper, we investigate the fractional-order Klein–Fock–Gordon equations on quantum dynamics using a new iterative method and residual power series method based on the Caputo operator. The fractional-order Klein–Fock–Gordon equation is a generalization of the traditional Klein–Fock–Gordon equation that allows for non-integer orders of differentiation. This equation has been used in the study of quantum dynamics to model the behavior of particles with fractional spin. The Laplace transform is employed to transform the equations into a simpler form, and the resulting equations are then solved using the proposed methods. The accuracy and efficiency of the method are demonstrated through numerical simulations, which show that the method is superior to existing numerical methods in terms of accuracy and computational time. The proposed method is applicable to a wide range of fractional-order differential equations, and it is expected to find applications in various areas of science and engineering.

Linearization criteria for two-dimensional systems of second-order ordinary differential equations (ODEs) have been derived earlier using complex symmetry analysis. For such systems, the linearizable form, linearization criteria and symmetry group classification are presented. In this paper, we extend the complex approach to obtain a complex-linearizable form of two-dimensional systems of third-order ODEs. This form leads to a linearizable class and linearization criteria of these systems of ODEs.

Whereas Lie had linearized scalar second order ordinary differential equations (ODEs) by point transformations, and later Chern had extended to the third order by using contact transformation, till recently no work had been done for higher order (or systems) of ODEs. Lie had found a unique class defined by the number of infinitesimal symmetry generators but the more general ODEs were not so classified. Recently, classifications of higher order and systems of ODEs were provided. In this paper we relate contact symmetries of scalar ODEs with point symmetries of reduced systems. We define a new type of transformation that builds upon this relation and obtain equivalence classes of scalar third order ODEs linearizable via these transformations. Four equivalence classes of such equations are seen to exist.

The present research correlates with a fuzzy hybrid approach merged with a new iterative transform method known as the fuzzy new iterative transform method. With the help of Atangana-Baleanu under generalized Hukuhara differentiability, we show that this system works well by getting fractional fuzzy Cauchy reaction-diffusion equations with the initial fuzzy condition. Fractional Cauchy reaction-diffusion equations play a significant role in diffusion, and instabilities can lead to formation and stabilization. The suggested technique looks at fuzzy set theory to figure out how to solve the fuzzy Cauchy reaction-diffusion equations. In this way, the components can be quickly defined and a couple of numerical solutions with the uncertainty parameter can be found. Several numerical instances are looked at to show how effective and valuable the proposed technique is to see if the given problem will come to a solution. The simulation results show that the fuzzy new iterative transform method is an excellent way to study a proposed models behaviour precisely and accurately.