Laguerre Wavelets Exact Parseval Frame-based Numerical Method for the Solution of System of Differential Equations
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Laguerre Wavelets Exact Parseval Frame-based Numerical Method for the Solution of System of Differential Equations S. C. Shiralashetti1 · S. Kumbinarasaiah2 © Springer Nature India Private Limited 2020
Abstract In this study, the Laguerre wavelets exact Parseval frame is introduced and proposed an effective numerical algorithm to get a numerical solution for the system of differential equations based on the Laguerre wavelets exact Parseval frame. This algorithm includes the collocation method and truncated Laguerre wavelet frames. Here, we reduce the system of differential equations into a set of algebraic equations which are having unknown Laguerre wavelet frame coefficients. Some numerical examples are given and compared to the numerical solution by the present method with the Adomian decomposition method. Moreover, the modeling of the spreading of a non-fatal disease in a population, which represents a system of an ordinary differential equation is numerically solved by the proposed technique and compared with the Adomina decomposed method. The obtained results reveal that the present algorithm provides a good approximation than existing methods. Keywords Laguerre wavelets · Exact Parseval frame · Collocation method · The system of differential equations Mathematics Subject Classification 65T60 · 42C15 · 93C15
Introduction Wavelets are special functions in a limited domain that is, a wave function instead of oscillating forever it drops to zero. Recently, we are facing different kinds of wavelets which are depending on two parameters such as, n is dilation parameter and k is the translation parameter [20]. The theory and application of wavelets is a comparatively young branch in signal processing and mathematical field. It has been applied in engineering disciplines, such as signal analysis, time–frequency analysis, and engineering mathematics [21–28]. In this study, we proposed a new algorithm to obtain numerical solutions for the system of ordinary differential equations with different constraints. It is very important to obtain
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S. Kumbinarasaiah [email protected]
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Department of Mathematics, Karnatak University, Dharwad, Karnataka 580003, India
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Department of Mathematics, Bangalore University, Bengaluru, Karnataka 560056, India 0123456789().: V,-vol
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numerical solutions for such a system of nonlinear ordinary differential equations in many different fields of science and engineering such as chemical physics, fluid mechanics, solid-state physics, plasma physics, and plasma waves. Most realistic systems of ordinary differential equations don’t have exact solutions, therefore, we need numerical techniques [12]. Consider the system of an ordinary differential equation is of the form: n y dp1 (x) f x, y1n 1 , y2n 2 , . . . , y p p , (1) where d ≥ n i ∈ {0}U N , p1 and i 1, . . . , p, p is any natural number, d and n i represents the order of the derivatives. Corresponding initial conditions are as follows, y dp11 (a1 ) b
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