Non-probabilistic Solutions of Uncertain Fractional Order Diffusion Equations

Abstract

This paper investigates the numerical solution of uncertain fractional order diffusion equation subject to various external forces. Homotopy Perturbation Method (HPM) is used for the analysis. Uncertainties present in the system are modelled through triangular convex normalised fuzzy sets. A new computational technique has been proposed based on double parametric form of fuzzy numbers to handle the fuzzy fractional diffusion equation. Using the single parametric form of fuzzy numbers, the original fuzzy fractional diffusion equation is converted first to an interval fuzzy fractional differential equation. Next this equation is transformed to crisp form by applying double parametric form of fuzzy numbers. Finally the same is solved by HPM symbolically to obtain the uncertain bounds of the solution. Obtained results are depicted in term of plots. Results obtained by the proposed method are compared with existing results in special cases. calculus in the last few decades. Several excellent books related to this have also been written by different authors representing the scope and various aspects of fractional calculus such as in [30,38,[47][48][49]53]. These books also give an extensive review on fractional derivative and its applications which may help the reader for understating the basic concepts of fractional calculus and its application. As regards, many authors have developed various methods to solve fractional differential and integral equations of physical systems. Different numerical methods and applications of fractional differential equations are also introduced in [1, 13,18,24,33,36,40,42,45,54,62,65].In particular fractional order diffusion equations have been analysed by various authors [9, 12, 14-16, 31-35, 37, 43, 46, 51] due to the great importance in many areas of science and engineering. In general the system parameters and variables involved in the diffusion processes are considered as crisp or defined exactly. But in actual practice, rather than the particular value, only uncertain or vague estimates about the variables and parameters are known. Because those are found in general by some observation, experiment or experience. So, to handle these uncertainties and vagueness, one may use fuzzy parameters and variables in the governing differential equations.Both uncertainty and fractional differential equations play a vital role in real life problems. Very recently few authors studied fractional differential equations with uncertainty. Some recent contributions on the theory of fuzzy differential equations and fuzzy fractional differential equations can be seen in [2-6, 8, 11, 26, 28, 29, 39, 44, 52, 55-58, 61]. The concept of fuzzy fractional differential equation was introduced recently by Agrawal et al. [3]. Arshad and Lupulescu [6] proved some results on the existence and uniqueness of solutions of fuzzy fractional differential equations based on the concept of fuzzy differential equations of fractional order introduced by Agrawal [3]. Arshad and Lupulescu [5] investigated the fractional dif...

Solution of interval shallow water wave equations using homotopy perturbation method

“…Behera and Chakraverty (2015) recently proposed a single parametric concept to solve fully interval/fuzzy system of linear equation. Furthermore, Tapaswini and Chakraverty (2014a, 2014b, 2013) used homotopy perturbation method (HPM) for finding the solution of uncertain fractional order diffusion equation and nth-order interval/fuzzy differential equations. A midpoint-based approach for finding the solution of nth-order interval differential equation was also proposed by Tapaswini and Chakraverty (2014a, 2014b).…”