In this article, we solve nonlinear systems of third order KdV Equations and the systems of coupled Burgers equations in one and two dimensions with the help of two different methods. The suggested techniques in addition with Laplace transform and Atangana–Baleanu fractional derivative operator are implemented to solve four systems. The obtained results by implementing the proposed methods are compared with exact solution. The convergence of the method is successfully presented and mathematically proved. The results we get are compared with exact solution through graphs and tables which confirms the effectiveness of the suggested techniques. In addition, the results obtained by employing the proposed approaches at different fractional orders are compared, confirming that as the value goes from fractional order to integer order, the result gets closer to the exact solution. Moreover, suggested techniques are interesting, easy, and highly accurate which confirm that these methods are suitable methods for solving any partial differential equations or systems of partial differential equations as well.

Motivated by the importance of diffusion equations in many physical situations in general and in plasma physics in particular, therefore, in this study, we try to find some novel solutions to fractional-order diffusion equations to explain many of the ambiguities about the phenomena in plasma physics and many other fields. In this article, we implement two well-known analytical methods for the solution of diffusion equations. We suggest the modified form of homotopy perturbation method and Adomian decomposition methods using Jafari-Yang transform. Furthermore, illustrative examples are introduced to show the accuracy of the proposed methods. It is observed that the proposed method solution has the desire rate of convergence toward the exact solution. The suggested method’s main advantage is less number of calculations. The proposed methods give series form solution which converges quickly towards the exact solution. To show the reliability of the proposed method, we present some graphical representations of the exact and analytical results, which are in strong agreement with each other. The results we showed through graphs and tables for different fractional-order confirm that the results converge towards exact solution as the fractional-order tends towards integer-order. Moreover, it can solve physical problems having fractional order in different areas of applied sciences. Also, the proposed method helps many plasma physicists in modeling several nonlinear structures such as solitons, shocks, and rogue waves in different plasma systems.

The current article discusses the fuzzy new iterative transform approach, which is a combination of a fuzzy hybrid methodology and an iterative transformation technique. We establish the consistence of our strategy by obtaining fractional fuzzy Helmholtz equations with the initial fuzzy condition using the Caputo derivative under generalized Hukuhara differentiability. The series obtained result was calculated and compared to the proposed equations of the actual result. Three challenges were provided to validate our method, and the outcomes were approximated in fuzzy form. In each of the three examples, the upper and bottom halves of the fuzzy solution were approximated utilizing two various fractional order between 0 and 1. Due to the fact that it globalizes the dynamical behavior of the specified equation, it produces all forms of fuzzy results at any fractional order between 0 and 1. Due to the fact that the fuzzy numbers presents the result in a lower and upper branches fuzzy type, the unknown quantity incorporates fuzziness as well. It is critical to emphasize that the purpose of the proposed fuzziness approach is to demonstrate the efficiency and superiority of numerical solutions to nonlinear fractional fuzzy partial differential equations that arise in complex and physical structures.

This paper evaluates a semianalytical strategy combined with a novel fuzzy integral transformation and an iterative method inside the fuzziness concept known as the new iterative transform method. Additionally, we apply the abovementioned technique to the fractional fuzzy Kuramoto-Sivashinsky equations with g H -differentiability by employing various initial conditions. Numerous algebraic properties of the fuzzy fractional derivative Atangana-Baleanu operator are illustrated concerning the Shehu transformation to demonstrate their utility. Additionally, a general technique for Atangana-Baleanu fuzzy fractional derivatives is proposed in the sense of Caputo. It is important to note that the purpose of the suggested fuzziness technique is to establish the efficiency and accuracy of analytical solution to nonlinear fuzzy fractional partial differential equations that emerge in complex and physical structures.

In this work, ZZ transformation is combined with the Adomian decomposition method to solve the dynamical system of fractional order. The derivative of fractional order is represented in the Atangana–Baleanu derivative. The numerical examples are combined for their approximate-analytical solution. It is explored using graphs that indicate that the actual and approximation results are close to each other, demonstrating the method’s usefulness. Fractional-order solutions are the most in line with the dynamics of the targeted problems, and they provide an endless number of options for an optimal mathematical model solution for a particular physical phenomenon. This analytical approach produces a series form solution that is quickly convergent to exact solutions. The acquired results suggest that the novel analytical solution technique is simple to use and very successful at assessing complicated problems that arise in related fields of research and technology.