Generalized Nonpolynomial Spline Method by Fractional Order

Journal of the Association of Arab Universities for Basic and Applied Sciences

The spline collocation method is a competent and highly effective mathematical tool for constructing the approximate solutions of boundary value problems arising in science, engineering and mathematical physics. In this paper, a quintic polynomial spline collocation method is employed for a class of fractional boundary value problems (FBVPs). The FBVPs are expressed in terms of Caputo's fractional derivative in this approach. The consistency relations are derived in order to compute the approximate solutions of FBVPs. Finally, numerical results are given, which demonstrate the effectiveness of the numerical scheme.

Abstract and Applied Analysis, 2012

Suitable spline functions of polynomial form are derived and used to solve linear and nonlinear fractional differential equations. The proposed method is applicable for0<α≤1andα≥1, whereαdenotes the order of the fractional derivative in the Caputo sense. The results obtained are in good agreement with the exact analytical solutions and the numerical results presented elsewhere. Results also show that the technique introduced here is robust and easy to apply.

Mathematics and Statistics, 2023

Although there are theoretical conclusions about the existence, uniqueness, and properties of solutions to ordinary and partial differential equations, only the simplest and most straightforward particular problems can usually be solved explicitly, especially when nonlinear terms are involved, and we typically develop approximation. In order to resolve the form problem of fractional order beginning value (1) by lacunary interpolation with a fractional degree spline function, the main goal of this paper is to investigate and improve some approximate solutions as well as new approximate solution techniques that have been proposed for the first time. From a practical standpoint, the numerical solution of these differential equations is crucial because only a tiny portion of equations can be resolved analytically. For fractional differential equations that are sensitive to the beginning conditions, we provide a fractional spline approach. The polynomial coefficient-based spline interpolation must be constructed using the Caputo fractional integral and derivative. For the given spline function, a stability analysis is completed after investigating error boundaries. The numerical rationale for the suggested technique is thought to use three cases. The outcomes demonstrate how effective the spline fractional technique is in interpolating the coefficient with fractional polynomials. Finally, to demonstrate the effectiveness and correctness of the suggested strategy, general procedure programs are created in MATLAB and used to a number of instructive cases.

Approximation Theory XV: San Antonio 2016, 2017

We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivatives, i.e., the fractional-order logistic equation. The use of the fractional B-splines allows us to express the fractional derivatives of the approximating function in an analytical form. Thus, the fractional collocation method is easy to implement, accurate, and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.

Smoothing noisy data with spline functions is well known in approximation theory.

International Journal of Modeling, Simulation, and Scientific Computing, 2010

In this article, we introduce spline collocation methods to solve fractional boundary value problems (BVPs). The existence and uniqueness theorem of collocation solutions is studied, and the error estimate of collocation solutions is discussed. Numerical experiments are presented to demonstrate the theoretical results.

Fractal and Fractional, 2021

In many applications, real phenomena are modeled by differential problems having a time fractional derivative that depends on the history of the unknown function. For the numerical solution of time fractional differential equations, we propose a new method that combines spline quasi-interpolatory operators and collocation methods. We show that the method is convergent and reproduces polynomials of suitable degree. The numerical tests demonstrate the validity and applicability of the proposed method when used to solve linear time fractional differential equations.

In this paper, a new fractional spline function of polynomial form with the idea of the lacunary interpolation is considered to nd approximate solution for fractional di erential equations (FDEs). The proposed method is applicable for 2 (0; 1], where denotes the order of the fractional derivative in the Caputo sense. Convergence analysis of the method is considered. Some illustra- tive examples are presented and the obtained results reveal that the proposed technique is very e ective, convenient and quite accurate to such considered problems. The study is conducted through illustrative examples and error analysis.

Mathematics

This paper presents a groundbreaking numerical technique for solving nonlinear time fractional differential equations, combining the conformable continuity equation (CCE) with the Non-Polynomial Spline (NPS) interpolation to address complex mathematical challenges. By employing conformable descriptions of fractional derivatives within the CCE framework, our method ensures enhanced accuracy and robustness when dealing with fractional order equations. To validate our approach’s applicability and effectiveness, we conduct a comprehensive set of numerical examples and assess stability using the Fourier method. The proposed technique demonstrates unconditional stability within specific parameter ranges, ensuring reliable performance across diverse scenarios. The convergence order analysis reveals its efficiency in handling complex mathematical models. Graphical comparisons with analytical solutions substantiate the accuracy and efficacy of our approach, establishing it as a powerful tool...

International Journal of Computer Mathematics, 2014

In this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the fourparameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

AIP Conf. Proc. 2096, 020004-1–020004-9, 2019

Abstarct: In this paper, we introduce a generalized cubic spline method with fractional order and study with error optimality, convergence analysis and to find out an approximate solution for fractional differential equations. This method is proposed to be applicable for α ∈ (0,1) , where a denotes the order of the fractional derivative in the Caputo sense. Error bounded of this aim is chosen and discussed, some explained example are demonstrated and also it is compared for each example that value of is changed for some values

This paper presents a formulation and a study of three interpolatory fractional splines these are in the class of m, m = 2, 4, 6,  = 0:5. We extend fractional splines function with uniform knots to approximate the solution of fractional equations. The developed of spline method is to analysis convergence fractional order derivatives and estimating error bounds. We propose spline fractional method to solve fractional differentiation equations. Numerical example is given to illustrate the applicability and accuracy of the methods.

Chaos, Solitons & Fractals, 2017

This paper develops a technique for the approximate solution of a class of variable-order fractional differential equations useful in the area of fluid dynamics. The method adopts a piecewise integro quadratic spline interpolation and is used in the study of the variable-order fractional Bagley-Torvik and Basset equations. The accuracy of the proposed algorithm is verified by means of illustrative examples.

Fractal and Fractional, 2022

Developing mathematical models of fractional order for physical phenomena and constructing numerical solutions for these models are crucial issues in mathematics, physics, and engineering. Higher order temporal fractional evolution problems (EPs) with Caputo’s derivative (CD) are numerically solved using a sextic polynomial spline technique (SPST). These equations are frequently applied in a wide variety of real-world applications, such as strain gradient elasticity, phase separation in binary mixtures, and modelling of thin beams and plates, all of which are key parts of mechanical engineering. The SPST can be used for space discretization, whereas the backward Euler formula can be used for time discretization. For the temporal discretization, the method’s convergence and stability are assessed. To show the accuracy and applicability of the proposed technique, numerical simulations are employed.

Applied Mathematics and Computation, 2018

This paper is devoted to the application of the method of lines to solve one-dimensional diffusion equation where the classical (integer) second derivative is replaced by a fractional derivative of the Caputo type of order α less than 2 as the space derivative. A system of initial value problems approximates the solution of the fractional diffusion equation with spline approximation of the Caputo derivative. The result is a numerical approach of order O(x 2 + t m) , where x and t denote spatial and temporal step-sizes, and 1 ≤ m ≤ 5 is an integer which is set by an ODE integrator that we used. The convergence and numerical stability of the method are considered, and numerical tests to investigate the efficiency and feasibility of the scheme are provided.