Numerical Solution of Nonlinear Fractional Bratu Equation with Hybrid Method
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Numerical Solution of Nonlinear Fractional Bratu Equation with Hybrid Method P. Pirmohabbati1 · A. H. Refahi Sheikhani1
· A. Abdolahzadeh Ziabari2
Accepted: 7 October 2020 © Springer Nature India Private Limited 2020
Abstract This work, we studied the Bratu differential equation of the fractional-order, numerically via introducing a hybrid method. This method is a combination of the Chebyshev polynomials and the block-pulse wavelets matrix of fractional order integration concerning the Caputo sense. Our approach transforms the nonlinear differential equation into a nonlinear algebraic system. We analyzed various forms of the fractional Bratu equation with different parameters of the equation and its fractional derivative orders. Besides, we show that the introduced method is convergent. We present the obtained results in tables and graphs. Based on these results, we can see the accuracy and convergence of the solutions. Since the Bratu equation is nonlinear so, based on the results obtained, we can say that the used method in approximating the solutions has acceptable accuracy and performance. Keywords Fractional Bratu differential equation · Block-Pulse functions · Chebyshev polynomials · Hybrid method · Numerical approximation · Operational matrices Mathematics Subject Classification 26A33 · 97N40 · 34A08 · 35E15
Introduction A differential equation expresses the relationship between a function and its derivatives. They are unquestionably vital for modern science and engineering. Indeed, the explanation and interpretation of the behavior of natural phenomena are possible through the study of relevant mathematical models. In mathematics, a model is an explanation of an actual-world system
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A. H. Refahi Sheikhani [email protected]; [email protected] P. Pirmohabbati [email protected] A. Abdolahzadeh Ziabari [email protected]
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Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran
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Department of Physics, Faculty of Science, Lahijan Branch, Islamic Azad University, Lahijan, Iran 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
(2020) 6:162
based on mathematical language, logic, and concepts [11, 12, 36]. The differential equations are one of the essential tools in engineering and basic sciences such as physic, chemistry, geology, and so on [30, 42, 43]. It is easy to see that most of the phenomena in the natural world have non-linear quiddity. Therefore, in explaining and analyzing the demeanor of these phenomena, using non-linear differential equations looks unavoidable. So, the investigation of non-linear differential equations is essential in all areas of sciences and technology. The exact solutions to some of these nonlinear differential equations do not exist. Therefore, finding approximate solutions for these equations can play a vital role in the investigation of nonlinear physical phenomena [4, 9, 10, 50]. One of these nonlinear equations is Bratu’s equation named after G. Bratu, a Fr
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