Abstract
Recent advancement in the field of nonlinear analysis and fractional calculus help to address the rising challenges in the solution of nonlinear fractional partial differential equations. This paper presents a hybrid technique, a combination of Tarig transform and Projected Differential Transform Method (TPDTM) to solve nonlinear fractional partial differential equations. The effectiveness of the method is examined by solving three numerical examples that arise in the field of heat transfer analysis. In this proposed scheme, the solution is obtained as a convergent series and the result is used to analyze the hyper diffusive process with pre local information regarding the heat transfer for different values of fractional order. In order to validate the results, a comparative study has been carried out with the solution obtained from the two methods, the Laplace Adomian Decomposition Method (LADM) and Homotophy Pertubation Method (HPM) and the result thus observed coincided with each other. Inspite of the uniformity between the solutions, the proposed hybrid technique had to overcome the complexity of manupulation of Adomian polynomials and evaluation of integrals in LADM and HPM respectively. The methodology and the results presented in this paper clearly reveals the computational efficiency of the present method. Due to its computational efficiency, the TPDTM has the potential to be used as a novel tool not only for solving nonlinear fractional differential equations but also for analysing the prelocal information of the system.
Highlights
Linear and nonlinear fractional partial differential equations have attracted many researchers due to their enormous applications in various fields of engineering like vibration analysis, fluid flow and heat transfer analysis etc
The temperature distribution is predicted along a rod for a linear fractional time heat conduction problem (Example 1) by using the Tarig Projected differential transform method with the parameters κ = 1.158 and length of the rod l = 80 cm
The Tarig projected differential transform method (TPDTM) is applied to solve nonlinear fractional time heat conduction problem with variable thermal conductivity (Example 3) and the temperature distributions are presented for the case beta=0.05 and 0.10 when alpha=0.75
Summary
Linear and nonlinear fractional partial differential equations have attracted many researchers due to their enormous applications in various fields of engineering like vibration analysis, fluid flow and heat transfer analysis etc. Elzaki (2012) combined both Elzaki transform and differential transform method (DTM) and applied this hybrid technique to solve nonlinear partial differential equations. Elzaki (2015) applied projected differential transform method to solve nonlinear, space and time fractional partial differential equations. This paper proposes a hybrid technique to solve nonlinear fractional partial differential equations (NLFPDE) called Tarig projected differential transform method (TPDTM). This method has been applied to solve linear, nonlinear fractional partial differential equations in heat transfer analysis and the temperature distribution functions are obtained as a convergent series for different values of fractional order and nonlinearity.
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