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Keywords = quintic Hermite spline

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17 pages, 532 KB  
Article
Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines
by Priyanka Priyanka, Shelly Arora, Saroj Sahani and Sharandeep Singh
Math. Comput. Appl. 2024, 29(6), 100; https://doi.org/10.3390/mca29060100 - 2 Nov 2024
Viewed by 1075
Abstract
Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and [...] Read more.
Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique. Full article
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22 pages, 844 KB  
Review
A Review of Collocation Approximations to Solutions of Differential Equations
by Pravin Singh, Nabendra Parumasur and Shivani Singh
Mathematics 2022, 10(23), 4438; https://doi.org/10.3390/math10234438 - 24 Nov 2022
Cited by 3 | Viewed by 3241
Abstract
This review considers piecewise polynomial functions, that have long been known to be a useful and versatile tool in numerical analysis, for solving problems which have solutions with irregular features, such as steep gradients and oscillatory behaviour. Examples of piecewise polynomial functions used [...] Read more.
This review considers piecewise polynomial functions, that have long been known to be a useful and versatile tool in numerical analysis, for solving problems which have solutions with irregular features, such as steep gradients and oscillatory behaviour. Examples of piecewise polynomial functions used include splines, in particular B-splines, and Hermite functions. Spline functions are useful for obtaining global approximations whilst Hermite functions are useful for approximation over finite elements. Our aim in this review is to study quintic Hermite functions and develop a numerical collocation scheme for solving ODEs and PDEs. This choice of basis functions is further motivated by the fact that we are interested in solving problems having solutions with steep gradients and oscillatory properties, for which this approximation basis seems to be a suitable choice. We derive the quintic Hermite basis and use it to formulate the orthogonal collocation on finite element (OCFE) method. We present the error analysis for third order ODEs and derive both global and nodal error bounds to illustrate the super-convergence property at the nodes. Numerical simulations using the Julia programming language are performed for both ODEs and PDEs and enhance the theoretical results. Full article
(This article belongs to the Section E: Applied Mathematics)
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