In this study, we applied the Laplace residual power series method (LRPSM) to expand the solution of the nonlinear time-fractional coupled Hirota–Satsuma and KdV equations in the form of a rapidly convergent series while considering Caputo fractional derivatives. We demonstrate the applicability and accuracy of the proposed method with some examples. The numerical results and the graphical representations reveal that the proposed method performs extremely well in terms of efficiency and simplicity. Therefore, it can be utilized to solve more problems in the field of non-linear fractional differential equations. To show the validity of the proposed method, we present a numerical application, compute two kinds of errors, and sketch figures of the obtained results.

Fractional-order boundary value problems are used to model certain phenomena in chemistry, physics, biology, and engineering. However, some of these models do not meet the existence and uniqueness required in the mainstream of mathematical processes. Therefore, in this paper, the existence, stability, and uniqueness for the solution of the coupled system of the Caputo-type sequential fractional differential equation, involving integral boundary conditions, was discussed, and investigated. Leray–Schauder’s alternative was applied to derive the existence of the solution, while Banach’s contraction principle was used to examine the uniqueness of the solution. Moreover, Ulam–Hyers stability of the presented system was investigated. It was found that the theoretical-related aspects (existence, uniqueness, and stability) that were examined for the governing system were satisfactory. Finally, an example was given to illustrate and examine certain related aspects.

This study presents a novel triple integral transformation, called the double ARA–generalized Laplace transform (DAGLT), and applies it to solve (2 + 1)–dimensional singular pseudoparabolic equations in both linear and nonlinear forms. The work begins by outlining the fundamental definitions, theorems, and characteristics of the single ARA and generalized Laplace transforms, then proceeds to unify them into a single framework (DAGLT). Detailed derivations of the core properties linearity, shifting, and partial derivative transforms demonstrate the theoretical soundness and broad applicability of DAGLT. Utilizing the Adomian decomposition method merged with this new transform, the paper presents an efficient procedure for obtaining closed‐form or series‐based solutions to challenging singular pseudoparabolic partial differential equations. Several illustrative examples showcase the effectiveness of the proposed technique, confirming that its solutions align with known results in the literature. The main advantages of the method are that it can handle singularities, and it improves the convergence. This study is promising for dealing with different models of multidimensional partial differential equations that arise in physical science and models.

This paper successfully employs a combined methodology that integrates the reduced differential transform approach, Laplace transform, and Padé approximants to solve diverse partial differential equations with real and complex variables. The proposed method, known as the modified reduced differential transform method (MRDTM), extends the interval of convergence with less computing time. An interesting aspect of this method is its capability to produce an analytic exact solution with only a few computable terms. The paper provides practical applications through notable examples encompassing the Klein–Gordon equation, the Schrödinger equation, the nonlinear reaction–diffusion–convection equation, and systems of linear and nonlinear PDEs. Primary results on certain test problems demonstrate the efficiency and ability of the method in solving diverse classes of partial differential equations, and therefore, it can be used as an alternative for dealing with such problems that do not have analytic solutions.