Balarampur College

In this paper, a comparative study between two analytical methods: the Residual Power Series Transform Method (RPSTM) and the Optimal Homotopy Asymptotic Transform Method (OHATM) is conducted. The RPSTM combines the Residual Power Series Method with the Laplace Transform, while the OHATM integrates the Optimal Homotopy Asymptotic Method with it. Both methods are applied to time-fractional Cauchy reaction-diffusion equations with uncertainty, where the fractional derivative is considered in the Caputo sense. Approximate solutions, specifically the upper and lower bound solutions of the equations, are obtained by both RPSTM and OHATM. The study includes theoretical analysis, graphical illustrations, and numerical evaluations. The effectiveness and accuracy of both methods are demonstrated by the comparisons between the exact solutions and the approximate solutions in a fuzzy environment. It is revealed through the findings that the Residual Power Series Transform Method is simpler, more expedient, and more effective in obtaining approximate solutions of time-fractional Cauchy reaction-diffusion equations compared with the Optimal Homotopy Asymptotic Transform Method.

Integrated thermo-, photo-, and electrocatalysis, along with photothermal, electrothermal, and photo-electrochemical routes, enable efficient CO 2 -to-CH 4 conversion. Advanced catalyst design drives sustainable carbon recycling and energy storage.

In this article, we obtain the numerical solution of the fuzzy form of the fractional Sharma-Tasso-Olver equation using the Homotopy Analysis Transform Method. This method combines two powerful and well-known methods: the Homotopy Analysis Method and the Laplace Transform Method. Two approximate solutions of the fuzzy fractional Sharma-Tasso-Olver equation are obtained using this approach. Comparisons are made between the results obtained by the proposed method and the exact solution available in the open literature. All the obtained numerical computations justify that the proposed method is highly reliable, simple, efficient, and effective for handling fuzzy fractional-order equations like the Sharma-Tasso-Olver equation.

In this work, we extend the modified homotopy analysis transform method (MHATM) for studying the three different coupled time-fractional physical problems. The first one is coupled time-fractional Whitham-Broer-Kaup (W-B-K) equations, and others are coupled modified Boussinesq equations and coupled approximate long wave equations as the special cases of W-B-K equations. The fractional W-B-K model is a coupled structure that describes the nonlinear evolution of shallow-water waves. The novelty of the proposed algorithm, the fractional derivative, is taken in the Caputo-Fabrizio (CF) sense, which consists of an exponential form of the non-singular kernel. With the aid of Banach's fixed point theory, the uniqueness and convergence analysis for the coupled W-B-K equations is presented through the theorems. With the help of Picard's stable approach, the stability analysis of the proposed technique is shown. We study the comparison for the solutions of MHATM for CF time-fractional derivative with the solutions derived with the aid of other techniques. The main advantage of the MHATM with the assistance of CF derivative is that, it offers solutions to problems in a rapidly convergent series leading to ideal solutions. The accuracy and efficiency of the present method have been shown through different graphical as well as tabulated analyses. However, the results indicate that MHATM with CF fractional derivative is a good organization and applicable to solve highly nonlinear various fractional physical problems like W-B-K equations.

We have investigated the impact of annealing temperature on the dielectric properties of SmCrO_3 synthesized at two different annealing temperatures: 1073 K and 1673 K, resulting in samples with average grain sizes of approximately 150 nm (S150) 350 nm (S350), respectively. X-ray diffraction patterns confirm the single-phase orthorhombic structure with the Pnma (no. 62) space group. Temperature and frequency-dependent dielectric analysis show a prominent dependence of the dielectric behavior on annealing temperature. The S150 samples exhibited lower dielectric permittivity and dielectric loss compared to S350 samples. Interestingly, a distinctive step-like anomaly has been observed in the dielectric response of the S150 sample below 100 K. Activation energy values, obtained from Arrhenius fits, consistently fall within the 0.2 eV to 0.4 eV range above 100 for both samples, indicating a similar relaxation mechanism. However, near the anomaly at 100 K, a lower activation energy of 0.06 eV suggested a distinct relaxation mechanism. Nyquist plots are utilized to model the dielectric behaviour with equivalent circuits, providing insights into the underlying mechanisms. This study provides the intricate nature of the dielectric properties of SmCrO_3 and highlights the significant impact of annealing temperature on these properties.