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Keywords = fractional Riccati differential equation

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20 pages, 11438 KiB  
Article
Investigating Chaotic Techniques and Wave Profiles with Parametric Effects in a Fourth-Order Nonlinear Fractional Dynamical Equation
by Jan Muhammad, Ali H. Tedjani, Ejaz Hussain and Usman Younas
Fractal Fract. 2025, 9(8), 487; https://doi.org/10.3390/fractalfract9080487 - 24 Jul 2025
Viewed by 297
Abstract
In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the [...] Read more.
In this article, we investigate the fractional soliton solutions as well as the chaotic analysis of the fourth-order nonlinear Ablowitz–Kaup–Newell–Segur wave equation. This model is considered an intriguing high-order nonlinear partial differential equation that integrates additional spatial and dispersive effects to extend the concepts to more intricate wave dynamics, relevant in engineering and science for understanding complex phenomena. To examine the solitary wave solutions of the proposed model, we employ sophisticated analytical techniques, including the generalized projective Riccati equation method, the new improved generalized exponential rational function method, and the modified F-expansion method, along with mathematical simulations, to obtain a deeper insight into wave propagation. To explore desirable soliton solutions, the nonlinear partial differential equation is converted into its respective ordinary differential equations by wave transforms utilizing β-fractional derivatives. Further, the solutions in the forms of bright, dark, singular, combined, and complex solitons are secured. Various physical parameter values and arrangements are employed to investigate the soliton solutions of the system. Variations in parameter values result in specific behaviors of the solutions, which we illustrate via various types of visualizations. Additionally, a key aspect of this research involves analyzing the chaotic behavior of the governing model. A perturbed version of the system is derived and then analyzed using chaos detection techniques such as power spectrum analysis, Poincaré return maps, and basin attractor visualization. The study of nonlinear dynamics reveals the system’s sensitivity to initial conditions and its dependence on time-decay effects. This indicates that the system exhibits chaotic behavior under perturbations, where even minor variations in the starting conditions can lead to drastically different outcomes as time progresses. Such behavior underscores the complexity and unpredictability inherent in the system, highlighting the importance of understanding its chaotic dynamics. This study evaluates the effectiveness of currently employed methodologies and elucidates the specific behaviors of the system’s nonlinear dynamics, thus providing new insights into the field of high-dimensional nonlinear scientific wave phenomena. The results demonstrate the effectiveness and versatility of the approach used to address complex nonlinear partial differential equations. Full article
(This article belongs to the Section Mathematical Physics)
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20 pages, 1737 KiB  
Article
Operator-Based Approach for the Construction of Solutions to (CD(1/n))k-Type Fractional-Order Differential Equations
by Inga Telksniene, Zenonas Navickas, Romas Marcinkevičius, Tadas Telksnys, Raimondas Čiegis and Minvydas Ragulskis
Mathematics 2025, 13(7), 1169; https://doi.org/10.3390/math13071169 - 2 Apr 2025
Viewed by 414
Abstract
A novel methodology for solving Caputo D(1/n)Ck-type fractional differential equations (FDEs), where the fractional differentiation order is k/n, is proposed. This approach uniquely utilizes fractional power series expansions to transform the original [...] Read more.
A novel methodology for solving Caputo D(1/n)Ck-type fractional differential equations (FDEs), where the fractional differentiation order is k/n, is proposed. This approach uniquely utilizes fractional power series expansions to transform the original FDE into a higher-order FDE of type D(1/n)Ckn. Significantly, this perfect FDE is then reduced to a k-th-order ordinary differential equation (ODE) of a special form, thereby allowing the problem to be addressed using established ODE techniques rather than direct fractional calculus methods. The effectiveness and applicability of this framework are demonstrated by its application to the fractional Riccati-type differential equation. Full article
(This article belongs to the Special Issue Fractional-Order Systems: Control, Modeling and Applications)
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17 pages, 270 KiB  
Article
On the Oscillatory Behavior of a Class of Mixed Fractional-Order Nonlinear Differential Equations
by George E. Chatzarakis, N. Nagajothi, M. Deepa and Vadivel Sadhasivam
Symmetry 2025, 17(3), 446; https://doi.org/10.3390/sym17030446 - 17 Mar 2025
Viewed by 344
Abstract
This paper investigates the oscillatory behavior of a class of mixed fractional-order nonlinear differential equations incorporating both the Liouville right-sided and conformable fractional derivatives. Symmetry plays a key role in understanding the oscillatory behavior of these systems. The motivation behind this study arises [...] Read more.
This paper investigates the oscillatory behavior of a class of mixed fractional-order nonlinear differential equations incorporating both the Liouville right-sided and conformable fractional derivatives. Symmetry plays a key role in understanding the oscillatory behavior of these systems. The motivation behind this study arises from the need for a more generalized framework to analyze oscillatory behavior in fractional differential equations, bridging the gap in the existing literature. By employing the generalized Riccati technique and the integral averaging method, we establish new oscillation criteria that extend and refine previous results. Illustrative examples are provided to validate the theoretical findings and highlight the effectiveness of the proposed methods. Full article
(This article belongs to the Section Mathematics)
25 pages, 14310 KiB  
Article
A Robust and Versatile Numerical Framework for Modeling Complex Fractional Phenomena: Applications to Riccati and Lorenz Systems
by Waleed Mohammed Abdelfattah, Ola Ragb, Mohamed Salah and Mokhtar Mohamed
Fractal Fract. 2024, 8(11), 647; https://doi.org/10.3390/fractalfract8110647 - 6 Nov 2024
Cited by 2 | Viewed by 1001
Abstract
The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work [...] Read more.
The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10−8. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations. Full article
(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
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14 pages, 328 KiB  
Article
Oscillatory and Asymptotic Criteria for a Fifth-Order Fractional Difference Equation
by Qinghua Feng
Fractal Fract. 2024, 8(10), 590; https://doi.org/10.3390/fractalfract8100590 - 7 Oct 2024
Viewed by 940
Abstract
In this paper, using the properties of the conformable fractional difference and fractional sum, we initially establish some oscillatory and asymptotic criteria for a fifth-order fractional difference equation. Several critical inequalities, the Riccati transformation technique, and the integral technique are used in the [...] Read more.
In this paper, using the properties of the conformable fractional difference and fractional sum, we initially establish some oscillatory and asymptotic criteria for a fifth-order fractional difference equation. Several critical inequalities, the Riccati transformation technique, and the integral technique are used in the deduction process. We provide some example to test the results. The established criteria are new results in the study of oscillation, and can be extended to other types of high-order fractional difference equations as well as fractional differential equations with more complicated forms. Full article
(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
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22 pages, 2980 KiB  
Article
Approximate Solutions of Fractional Differential Equations Using Optimal q-Homotopy Analysis Method: A Case Study of Abel Differential Equations
by Süleyman Şengül, Zafer Bekiryazici and Mehmet Merdan
Fractal Fract. 2024, 8(9), 533; https://doi.org/10.3390/fractalfract8090533 - 11 Sep 2024
Viewed by 1297
Abstract
In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives [...] Read more.
In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. Full article
(This article belongs to the Special Issue Fractional Mathematical Modelling: Theory, Methods and Applications)
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23 pages, 6260 KiB  
Article
Interaction Solutions for the Fractional KdVSKR Equations in (1+1)-Dimension and (2+1)-Dimension
by Lihua Zhang, Zitong Zheng, Bo Shen, Gangwei Wang and Zhenli Wang
Fractal Fract. 2024, 8(9), 517; https://doi.org/10.3390/fractalfract8090517 - 30 Aug 2024
Viewed by 705
Abstract
We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite [...] Read more.
We extend two KdVSKR models to fractional KdVSKR models with the Caputo derivative. The KdVSKR equation in (2+1)-dimension, which is a recent extension of the KdVSKR equation in (1+1)-dimension, can model the soliton resonances in shallow water. Applying the Hirota bilinear method, finite symmetry group method, and consistent Riccati expansion method, many new interaction solutions have been derived. Soliton and elliptical function interplaying solution for the fractional KdVSKR model in (1+1)-dimension has been derived for the first time. For the fractional KdVSKR model in (2+1)-dimension, two-wave interaction solutions and three-wave interaction solutions, including dark-soliton-sine interaction solution, bright-soliton-elliptic interaction solution, and lump-hyperbolic-sine interaction solution, have been derived. The effect of the order γ on the dynamical behaviors of the solutions has been illustrated by figures. The three-wave interaction solution has not been studied in the current references. The novelty of this paper is that the finite symmetry group method is adopted to construct interaction solutions of fractional nonlinear systems. This research idea can be applied to other fractional differential equations. Full article
(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)
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13 pages, 617 KiB  
Article
Option Pricing with Fractional Stochastic Volatilities and Jumps
by Sumei Zhang, Hongquan Yong and Haiyang Xiao
Fractal Fract. 2023, 7(9), 680; https://doi.org/10.3390/fractalfract7090680 - 11 Sep 2023
Cited by 3 | Viewed by 2162
Abstract
Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities [...] Read more.
Empirical studies suggest that asset price fluctuations exhibit “long memory”, “volatility smile”, “volatility clustering” and asset prices present “jump”. To fit the above empirical characteristics of the market, this paper proposes a fractional stochastic volatility jump-diffusion model by combining two fractional stochastic volatilities with mixed-exponential jumps. The characteristic function of the log-return is expressed in terms of the solution of two-dimensional fractional Riccati equations of which closed-form solution does not exist. To obtain the explicit characteristic function, we approximate the pricing model by a semimartingale and convert fractional Riccati equations into a classic PDE. By the multi-dimensional Feynman-Kac theorem and the affine structure of the approximate model, we obtain the solution of the PDE with which the explicit characteristic function and its cumulants are derived. Based on the derived characteristic function and Fourier cosine series expansion, we obtain approximate European options prices. By differential evolution algorithm, we calibrate our approximate model and its two nested models to S&P 500 index options and obtain optimal parameter estimates of these models. Numerical results demonstrate the pricing method is fast and accurate. Empirical results demonstrate our approximate model fits the market best among the three models. Full article
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21 pages, 449 KiB  
Article
Hybrid GPU–CPU Efficient Implementation of a Parallel Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Riccati Equation of Fractional Variable Order
by Dmitrii Tverdyi and Roman Parovik
Mathematics 2023, 11(15), 3358; https://doi.org/10.3390/math11153358 - 31 Jul 2023
Cited by 6 | Viewed by 1706
Abstract
The numerical solution for fractional dynamics problems can create a high computational load, which makes it necessary to implement efficient algorithms for their solution. The main contribution to the computational load of such computations is created by heredity (memory), which is determined by [...] Read more.
The numerical solution for fractional dynamics problems can create a high computational load, which makes it necessary to implement efficient algorithms for their solution. The main contribution to the computational load of such computations is created by heredity (memory), which is determined by the dependence of the current value of the solution function on previous values in the time interval. In terms of mathematics, the heredity here is described using a fractional differentiation operator in the Gerasimov–Caputo sense of variable order. As an example, we consider the Cauchy problem for the non-linear fractional Riccati equation with non-constant coefficients. An efficient parallel implementation algorithm has been proposed for the known sequential non-local explicit finite-difference numerical solution scheme. This implementation of the algorithm is a hybrid one, since it uses both GPU and CPU computational nodes. The program code of the parallel implementation of the algorithm is described in C and CUDA C languages, and is developed using OpenMP and CUDA hardware, as well as software architectures. This paper presents a study on the computational efficiency of the proposed parallel algorithm based on data from a series of computational experiments that were obtained using a computing server NVIDIA DGX STATION. The average computation time is analyzed in terms of the following: running time, acceleration, efficiency, and the cost of the algorithm. As a result, it is shown on test examples that the hybrid version of the numerical algorithm can give a significant performance increase of 3–5 times in comparison with both the sequential version of the algorithm and OpenMP implementation. Full article
(This article belongs to the Special Issue Computational Mathematics and Mathematical Modelling)
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23 pages, 735 KiB  
Article
The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order
by Lakhlifa Sadek, Ahmad Sami Bataineh, Hamad Talibi Alaoui and Ishak Hashim
Fractal Fract. 2023, 7(4), 302; https://doi.org/10.3390/fractalfract7040302 - 30 Mar 2023
Cited by 30 | Viewed by 2285
Abstract
We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by [...] Read more.
We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods. Full article
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20 pages, 1551 KiB  
Article
Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber
by Dmitrii Tverdyi, Evgeny Makarov and Roman Parovik
Mathematics 2023, 11(4), 850; https://doi.org/10.3390/math11040850 - 7 Feb 2023
Cited by 8 | Viewed by 1787
Abstract
Mathematical modeling is used to study the hereditary mechanism of the accumulation of radioactive radon gas in a chamber with gas-discharge counters at several observation points in Kamchatka. Continuous monitoring of variations in radon volumetric activity in order to identify anomalies in its [...] Read more.
Mathematical modeling is used to study the hereditary mechanism of the accumulation of radioactive radon gas in a chamber with gas-discharge counters at several observation points in Kamchatka. Continuous monitoring of variations in radon volumetric activity in order to identify anomalies in its values is one of the effective methods for studying the stress–strain state of the geo-environment with the possibility of building strong earthquake forecasts. The model equation of radon transfer, taking into account its accumulation in the chamber and the presence of the hereditary effect (heredity or memory), is a nonlinear differential Riccati equation with non-constant coefficients with a fractional derivative in the sense of Gerasimov–Caputo, for which local initial conditions are set (Cauchy problem). The proposed hereditary model of radon accumulation in the chamber is a generalization of the previously known model with an integer derivative (classical model). This fact indicates the preservation of the properties of the previously obtained solution according to the classical model, as well as the presence of new properties that are applied to the study of radon volumetric activity at observation points. The paper shows that due to the order of the fractional derivative, as well as the quadratic nonlinearity in the model equation, the results of numerical simulation give a better approximation of the experimental data of radon monitoring than by classical models. This indicates that the hereditary model of radon transport is more flexible, which allows using it to describe various anomalous effects in the values of radon volume activity. Full article
(This article belongs to the Section E2: Control Theory and Mechanics)
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17 pages, 497 KiB  
Article
Computational Analysis of the Fractional Riccati Differential Equation with Prabhakar-type Memory
by Jagdev Singh, Arpita Gupta and Devendra Kumar
Mathematics 2023, 11(3), 644; https://doi.org/10.3390/math11030644 - 27 Jan 2023
Cited by 17 | Viewed by 2035
Abstract
The key objective of the current work is to examine the behavior of the nonlinear fractional Riccati differential equation associated with the Caputo–Prabhakar derivative. An efficient computational scheme, that is, a mixture of homotopy analysis technique and sumudu transform, is used to solve [...] Read more.
The key objective of the current work is to examine the behavior of the nonlinear fractional Riccati differential equation associated with the Caputo–Prabhakar derivative. An efficient computational scheme, that is, a mixture of homotopy analysis technique and sumudu transform, is used to solve the nonlinear fractional Riccati differential equation. The convergence and uniqueness analysis for the solution of the implemented technique is shown. In addition, the numerical consequences are demonstrated in the form of graphical representations to verify the reliability of the applied method in obtaining the solution to the mathematical model with Prabhakar-type memory. Full article
(This article belongs to the Special Issue Fractional Partial Differential Equations: Theory and Applications)
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10 pages, 966 KiB  
Article
Constructing Analytical Solutions of the Fractional Riccati Differential Equations Using Laplace Residual Power Series Method
by Aliaa Burqan, Aref Sarhan and Rania Saadeh
Fractal Fract. 2023, 7(1), 14; https://doi.org/10.3390/fractalfract7010014 - 25 Dec 2022
Cited by 13 | Viewed by 2193
Abstract
In this article, a hybrid numerical technique combining the Laplace transform and residual power series method is used to construct a series solution of the nonlinear fractional Riccati differential equation in the sense of Caputo fractional derivative. The proposed method is implemented to [...] Read more.
In this article, a hybrid numerical technique combining the Laplace transform and residual power series method is used to construct a series solution of the nonlinear fractional Riccati differential equation in the sense of Caputo fractional derivative. The proposed method is implemented to construct analytical series solutions of the target equation. The method is tested for eminent examples and the obtained results demonstrate the accuracy and efficiency of this technique by comparing it with other numerical methods. Full article
(This article belongs to the Special Issue Advances in Fractional Differential Operators and Their Applications)
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21 pages, 1406 KiB  
Article
The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model
by Muhamad Deni Johansyah, Asep Kuswandi Supriatna, Endang Rusyaman and Jumadil Saputra
Mathematics 2022, 10(17), 3029; https://doi.org/10.3390/math10173029 - 23 Aug 2022
Cited by 9 | Viewed by 1942
Abstract
This work proposes and investigates the existence and uniqueness of solutions of Riccati Fractional Differential Equation (RFDE) with constant coefficients using Banach’s fixed point theorem. This theorem is the uniqueness theorem of a fixed point on a contraction mapping of a norm space. [...] Read more.
This work proposes and investigates the existence and uniqueness of solutions of Riccati Fractional Differential Equation (RFDE) with constant coefficients using Banach’s fixed point theorem. This theorem is the uniqueness theorem of a fixed point on a contraction mapping of a norm space. Furthermore, the combined theorem of the Adomian Decomposition Method (ADM) and Kamal’s Integral Transform (KIT) is used to convert the solution of the Fractional Differential Equation (FDE) into an infinite polynomial series. In addition, the terms of an infinite polynomial series can be decomposed using ADM, which assumes that a function can be decomposed into an infinite polynomial series and nonlinear operators can be decomposed into an Adomian polynomial series. The final result of this study is to find a solution of the RFDE approach to the economic growth model with a quadratic cost function using the combined ADM and KIT. The results showed that the RFDE solution on the economic growth model using the combined ADM and KIT showed a very good performance. Furthermore, the numerical solution of RFDE on the economic growth model is presented at the end of this work. Full article
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14 pages, 810 KiB  
Article
Enhancing the Accuracy of Solving Riccati Fractional Differential Equations
by Antonela Toma, Flavius Dragoi and Octavian Postavaru
Fractal Fract. 2022, 6(5), 275; https://doi.org/10.3390/fractalfract6050275 - 20 May 2022
Cited by 9 | Viewed by 2623
Abstract
In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with xα, with positive α. Fractional derivatives are in the Caputo sense. With the help of [...] Read more.
In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with xα, with positive α. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if α=1. Full article
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