On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burger’s equations using homotopy analysis transform method |
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| Institution: | 1. Department of Mathematics, Collage of Arts and Sciences, Najran University, Saudi Arabia;2. Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen;3. Department of Mathematics, Faculty of Sciences, Yarmouk University, Jordan;4. Department of Mathematics, Faculty of Sciences, Cankaya University, 06530, Balgat, Ankara, Turkey;5. Institute of Space Sciences, Magurele-Bucharest, Romania;6. CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México;1. Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada;2. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China;3. Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan;4. Faculty of Mathematical and Statistical Sciences, Shri Ramswaroop Memorial University, Deva Road, Lucknow 225003, Uttar Pradesh, India;5. Department of Applied Science and Humanities, Government Engineering College (Department of Science and Technology of the Government of Bihar), Newada 805122, Bihar, India;6. Department of Mathematics, JECRC University, Jaipur 303905, Rajasthan, India;7. Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India;8. Department of Mathematics, Faculty of Arts and Sciences, Cankaya University, Eskisehir, TR-06790 Etimesgut, Turkey;9. Institute of Space Sciences, Magurele-Bucharest, Romania;1. National College of Business Administration & Economics, Lahore, Pakistan;2. Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey |
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| Abstract: | In this paper we consider the homotopy analysis transform method (HATM) to solve the time fractional order Korteweg-de Vries (KdV) and Korteweg-de Vries-Burger’s (KdVB) equations. The HATM is a combination of the Laplace decomposition method (LDM) and the homotopy analysis method (HAM). The fractional derivatives are defined in the Caputo sense. This method gives the solution in the form of a rapidly convergent series with h-curves are used to determine the intervals of convergent. Averaged residual errors are used to find the optimal values of h. It is found that the optimal h accelerates the convergence of the HATM, with the rate of convergence depending on the parameters in the KdV and KdVB equations. The HATM solutions are compared with exact solutions and excellent agreement is found. |
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