An Efficient Approach to Fractional Analysis for Non-Linear Coupled Thermo-Elastic Systems
Qasim Khan
TL;DR
The paper tackles nonlinear coupled thermoelastic systems with fractional time dynamics by marrying the Aboodh transform with the Adomian decomposition method (ATDM) under Caputo derivatives. The methodology transforms fractional PDEs to a tractable algebraic form, derives recurrence relations for the Adomian components, and reconstructs the solution as a convergent series $u(x,t)=\sum_j u_j(x,t)$ and $v(x,t)=\sum_j v_j(x,t)$. It demonstrates through three illustrative examples that the fractional-order solutions converge to the integer-order solutions as $\beta$ approaches 1, and that increasing the number of Adomian terms enhances accuracy, with comparisons to LRPSM and LADM supporting the effectiveness of ATDM. The Maple-based algorithm and explicit closed-form-like expressions for low-order terms highlight the practical efficiency of the approach for nonlinear FPDEs in thermoelastic contexts, with potential extensions to broader nonlinear fractional systems.
Abstract
Nonlinear thermoelastic systems play a crucial role in understanding thermal conductivity, stresses, elasticity, and temperature interactions. This research focuses on finding solutions to these systems in their fractional forms, which is a significant aspect of the study. We consider various proposed models related to fractional thermoelasticity and derive results through sophisticated methodologies. Numerical simulations are conducted for both fractional and integer order thermoelastic coupled systems, with results presented in tables and graphs. The graphs indicate a close correspondence between the approximate and exact solutions. The solutions obtained demonstrate convergence for both fractional and integer order problems, ensuring accurate modeling. Furthermore, the tables confirm that greater accuracy can be achieved by increasing the number of terms in the series of solutions.