The discretization of fractional-order differential operators is the key to the digital realization of fractional-order controllers. This paper proposes an improved second-order fractional differential equation operation method based on power series expansion. The algorithm's operation speed and accuracy performance are analyzed. The research found that the algorithm proposed in this paper is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical model is unknown and the solution of linear systems.

#### Keywords

- second-order fractional differential equation
- fractional-order system
- recursive algorithm
- accuracy
- power series expansion

#### MSC 2010

- 34A08

For a long time, we have focused on the mathematical theory of fractional calculus (FOC). It was not until the mid and late 20th century that FOC theory was introduced and applied in some fields such as electrochemistry, rheology, signal processing, and power transmission theory. Recently, the research of fractional order systems has aroused the interest of many scholars [1]. In the field of automatic control, a new branch of fractional control has emerged.

The research of fractional order control should cover at least three aspects: (1) The establishment of a fractional-order system (FOS) model and its analysis based on describing the fractional-order object more accurately and concisely. (2) The fractional-order control strategy is selected to obtain better control performance. (3) Apply fractional arithmetic to process signals, data, etc. The above work is based on the calculation of FOC and FOS.

Based on the Tustin transformation theory and the analysis of the characteristics of the discrete generating function formula used for fractional operators, and using the Maclaurin series expansion of the binomial power function, this paper proposes an improved method based on the power series expansion (PSE) and the Tustin transformation. Fractional calculation method [2]. We apply it to the solution of the linear fractional system and derive the corresponding recurrence formula. The performance of the algorithm's operation speed and accuracy is analyzed through specific example simulations and comprehensive comparisons with several conventional methods. The algorithm proposed in this paper is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical model is unknown and the solution of linear systems.

The essence of the numerical method is to perform a discrete approximation of FDEs to obtain corresponding approximate numerical solutions [3]. The core problem is the discretization of the FOC operator. The basic idea of discretization is to use grid function ^{−1}) to approximate the FOC of function

Euler backward difference method

Tustin method (trapezoidal method or bilinear transformation)

Al-Alaoui method

Since the corresponding generating function is irrational, it is not convenient to be directly used for solving. We can approximate it rationally [4]. Commonly used methods are PSE and continued fraction expansion (CFE). The combination of the above-mentioned different methods can get different forms of FOC operator discrete algorithm (PSE-Tustin, CFE-Tustin). Of course, no matter whether it is the PSE method or the CFE method, the solution expression obtained is still an infinite number of items. In its practical application, it can be considered to approximate its finite term, that is, the short memory method (SMP).

The research-based on traditional integer-order systems is relatively mature, and it is a basic research idea and method to approximate FOSs with integer-order systems [5]. So the problem is transformed into using standard integer-order operators to approximate fractional-order operators. This also overcomes the difficulty of not allowing direct fractional operator operations to calculate and simulate existing software. The fractional operator ^{a}^{a}

In the research, it is found that the integral transformation formula of the analytical algorithm is complicated, the calculation amount is large, and the dimension of the state space method is often too high. Models after rational approximation are often inconvenient for the solution and further analysis because of their high order and complexity. The existing analytical algorithms are all aimed at special forms of equations, but many equations often do not have analytical solutions [6]. On the other hand, the application premise of the analytical method is that the prototype function is known, and the problem of calculus for the known data is often encountered in practical engineering applications. Such as system parameter identification, signal analysis, etc. When applying PSE or CFE to the Eqs. (2) to (4) to realize the discretization approximation, it is necessary to call the related functions in Matlab or Maple. The calculation speed is very slow, and the resulting form cannot be directly called in programming. But modern control is based on the computer as the main realization tool. The meaning of its ‘realization’ is not limited to the ‘realization’ of its analysis and design process, but more importantly, it is put into actual operation. There are obvious obstacles to the above methods.

It is well known that the mapping relationship between the ^{a} f

In the research based on the numerical example simulation, it is found that the Tustin method and Al-Alaoui method with the same number of expansion terms or orders are better than the Euler method in terms of accuracy no matter whether PSE or CFE is used. The Tustin method is slightly better than the Al-Alaoui method [8]. Therefore, from the perspective of response distortion or error, Tustin transform is a better choice.

The convergence of the algorithm is one of the important issues that must be considered. Due to the possible limitations of the algorithm itself, when it is applied to the approximation of fractional-order operators, the system's output will diverge due to improper selection of the order (Figure 1). The fundamental reason is that the algorithm cannot guarantee convergence, or the convergence is not good [9]. Theoretically, because FOC has a unique attenuation memory characteristic, it must be shown as the uniform attenuation of the weight coefficients of the power terms of ^{−1} when calculating. However, they are not all well-reflected when using Matlab or Maple to expand and approximate Eqs. (2) to (4) as a whole. Taking ^{−1} = 0, the coefficients of the first 50 terms of ^{−1} are shown in Figure 2. It can be seen that the coefficient oscillates and the jump degree is large, and when the finite term is approximated, it will inevitably lose the information with a large weight. But the combination of formula (2) and PSE is an exception (Figure 3), and the convergence speed is very fast. Another significant advantage of this combination is that the transformation formula is a binomial power function. The weight coefficients of the expansion of the power series can be obtained quickly and conveniently. This is difficult to achieve in other combinations.

Tustin transform formula (3) is a fractional power function, and its numerator and denominator have the same structure as formula (2). Suppose the numerator and denominator are approximated as polynomials, and the two are regarded as the numerator and denominator of the pulse transfer function of a filter. In that case, this filter is a discrete FOC filter [10]. Furthermore, we can use the Maclaurin series expansion of the binomial power function to ensure convergence, and the weight coefficient of the expansion term can be quickly and recursively obtained.

We carry out the Maclaurin series expansion of the numerator and denominator of the function

Based on the above analysis, this paper proposes an improved FOC algorithm based on PSE and Tustin transform and applies it to the solution of linear fractional systems.

We assume that the FOC of function ^{a} f^{−1})^{a} (take n terms to approximate) and write it in matrix form
^{a}f_{k}_{k}_{k}_{1}_{k}_{−1} + ⋯ + _{k}_{−1}_{1} + _{k}d f_{0} = (2/^{a} (_{k}_{1} _{k}_{−1} + ⋯ + _{k}_{−1} _{1} + _{k} f_{0}). Then

Step 1: Discrete _{k}

Step 2: The initial value of the parameter _{0} is generally controlled as _{0} = 0.

Step 3: Take _{1}. _{1} = (2/^{a}(_{1} + _{1} _{0}) − _{1}_{0}.

Step 4: Let _{2}: _{2} = (2/^{a} (_{2} + _{1} _{1} + _{2} _{0}) − (_{1}_{1} + _{2}_{0}).

Step 5: _{3}, _{4}, _{5},⋯ ,_{n}

First consider the general form of the differential equation of a special fractional-order linear system:
^{ai}y_{k}^{ai}y^{(ai)} and ^{(ai)} can be directly recursively obtained by formula (8) and formula (10). We substitute it into Eq. (12) and arrange it as

Step 1: Calculate the sequence _{k}

Step 2: We take the initial value: _{0} = 0, ^{ai}_{0} = 0.

Step 3: We make _{1} from Eq. (14). Substitute formula (13) to obtain the series ^{ai}_{1}.

Step 4: Let _{2} from Eq. (14), and substitute Eq. (13) to get the series ^{ai}_{2}.

Step 5: Use the results of the previous steps to calculate _{3}, _{4}, _{5},_{k}

First, to verify the effectiveness of the FOC recursive solution algorithm proposed in this paper, here is a simple simulation curve of the 0.5-order derivative, 0.5-order integral, and the first-order integral of the sine function

It can be seen that the algorithm curve in this paper is the same as the result of the conventional PSE expansion approximation of the Tustin transform operator as a whole. This is almost the same as the curve obtained by the most direct and simple G-L definition discrete method [11]. The G-L method is almost identical to the theory obtained by applying the Cauchy definition to remove the area around ^{+}. Therefore, the results in Figure 5 show that the algorithm in this paper is feasible.

The unit step response is obtained through a specific FOS model. Based on the accuracy and speed, a comprehensive analysis and comparison between the method in this paper and the commonly used algorithms are made. The calculation and simulation tool is Matlab7. Let the FDEs of the simulation example be
_{k}_{k}^{0.5}) respectively. Take the error function
^{*}/^{*} is the reference value. Here we take the

Algorithm error and accuracy indicators

0.0649 | 13.3 | ||

0.0301 | 28.6 | ||

0.0161 | 53.5 | ||

Full memory | 0.012 | 370.1 | |

0.862 | 1 | ||

0.0557 | 15.4 | ||

0.0194 | 44.5 | ||

0.0052 | 165.8 | ||

50 | 0.0683 | 12.6 | |

100 | 0.0397 | 21.7 | |

200 | 0.0123 | 70.1 | |

50 | 0.0575 | 15 | |

100 | 0.0543 | 15.9 | |

200 | 0.0536 | 16.1 |

PSE, power series expansion

The error is larger for step

Figure 6 shows that using the recursive discrete approximation algorithm given in this article to obtain the system's step response can obtain a relatively satisfactory result. The response curve of the memory length

It can be seen from Figure 8 that the initial response of the comparison curve between the algorithm in this paper and the conventional PSE-Tustin method is not much different. After 5 s, the difference is obvious, but this phenomenon is not reflected in Figure 5. The reason may be that different FOC discrete approximation methods exhibit different accuracy due to the difference in signal behavior. For this reason, we apply the fractional-order Al-Alaoui discrete operator to this example using the conventional PSE method and compare and analyze the results to show (Figure 9 and Table 1). The response curve shape of PSE-Al-Alaoui is similar to that of PSE-Tustin. In addition, the difference between the number of expansion terms of PSE-Tustin and the obvious influence on the accuracy is that the number of expansion terms has a less obvious influence on the accuracy. This shows that the convergence of the algorithm is better than that of PSE-Tustin. Overall, the results of the algorithm in this paper are the best. The PSE-Tustin method is better than the PSE-Al-Alaoui method.

This paper proposes an improved PSE-Tustin recursive algorithm. This algorithm is suitable for the fractional operation of arbitrary signals, including discrete data sequences whose mathematical models are unknown and the solution of linear systems. A simulation example verifies the effectiveness of the algorithm. We show the characteristics and advantages of the algorithm in this paper by comparing the accuracy with several commonly used fractional-order algorithms.

#### Algorithm error and accuracy indicators

0.0649 | 13.3 | ||

0.0301 | 28.6 | ||

0.0161 | 53.5 | ||

Full memory | 0.012 | 370.1 | |

0.862 | 1 | ||

0.0557 | 15.4 | ||

0.0194 | 44.5 | ||

0.0052 | 165.8 | ||

50 | 0.0683 | 12.6 | |

100 | 0.0397 | 21.7 | |

200 | 0.0123 | 70.1 | |

50 | 0.0575 | 15 | |

100 | 0.0543 | 15.9 | |

200 | 0.0536 | 16.1 |

Law of interest rate changes in financial markets based on the differential equation model of liquidity Basalt fibre continuous reinforcement composite pavement reinforcement design based on finite element model Industrial transfer and regional economy coordination based on multiple regression model Satisfactory consistency judgement and inconsistency adjustment of linguistic judgement matrix Spatial–temporal graph neural network based on node attention A contrastive study on the production of double vowels in Mandarin Research of cascade averaging control in hydraulic equilibrium regulation of heating pipe network Mathematical analysis of civil litigation and empirical research of corporate governance Health monitoring of Bridges based on multifractal theory Health status diagnosis of the bridges based on multi-fractal de-trend fluctuation analysis Performance evaluation of college laboratories based on fusion of decision tree and BP neural network Application and risk assessment of the energy performance contracting model in energy conservation of public buildings Sensitivity analysis of design parameters of envelope enclosure performance in the dry-hot and dry-cold areas The Spatial Form of Digital Nonlinear Landscape Architecture Design Based on Computer Big Data Analysis of the relationship between industrial agglomeration and regional economic growth based on the multi-objective optimisation model Constraint effect of enterprise productivity based on constrained form variational computing The impact of urban expansion in Beijing and Metropolitan Area urban heat Island from 1999 to 2019 TOPSIS missile target selection method supported by the posterior probability of target recognition Ultrasonic wave promoting ice melt in ice storage tank based on polynomial fitting calculation model The incentive contract of subject librarians in university library under the non-linear task importance Application of Fuzzy Mathematics Calculation in Quantitative Evaluation of Students’ Performance of Basketball Jump Shot Visual error correction of continuous aerobics action images based on graph difference function Application of Higher Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Application of Forced Modulation Function Mathematical Model in the Characteristic Research of Reflective Intensity Fibre Sensors Radioactive source search problem and optimisation model based on meta-heuristic algorithm Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East Corporate social responsibility fulfilment, product-market competition and debt risk: Evidence from China ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Reconstruction of multimodal aesthetic critical discourse analysis framework Image design and interaction technology based on Fourier inverse transform What does students’ experience of e-portfolios suggest Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity Data mining of Chain convenience stores location The optimal model of employment and entrepreneurship models in colleges and universities based on probability theory and statistics A generative design method of building layout generated by path Parameter Id of Metal Hi-pressure State Equation Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control Intelligent Recommendation System for English Vocabulary Learning – Based on Crowdsensing Regarding new wave distributions of the non-linear integro-partial Ito differential and fifth-order integrable equations Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Application of regression function model based on panel data in bank resource allocation financial risk management Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Badminton players’ trajectory under numerical calculation method Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management Exploration on the collaborative relationship between government, industry, and university from the perspective of collaborative innovation The impact of financial repression on manufacturing upgrade based on fractional Fourier transform and probability AtanK-A New SVM Kernel for Classification Validity and reliability analysis of the Chinese version of planned happenstance career inventory based on mathematical statistics Visual positioning system for marine industrial robot assembly based on complex variable function Mechanical behaviour of continuous girder bridge with corrugated steel webs constructed by RW Research on the influencing factors of agricultural product purchase willingness in social e-commerce situation Study of a linear-physical-programming-based approach for web service selection under uncertain service quality A mathematical model of plasmid-carried antibiotic resistance transmission in two types of cells Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Innovations to Attribute Reduction of Covering Decision System Based on Conditional Information Entropy Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Adoption of deep learning Markov model combined with copula function in portfolio risk measurement Good congruences on weakly U-abundant semigroups Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Mathematical simulation analysis of optimal detection of shot-putters’ best path Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Nonlinear Differential Equations in the Teaching Model of Educational Informatisation Fed-UserPro: A user profile construction method based on federated learning The evaluation of college students’ innovation and entrepreneurship ability based on nonlinear model Smart Communities to Reduce Earthquake Damage: A Case Study in Xinheyuan, China Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations Institutional investor company social responsibility report and company performance Mathematical analysis of China's birth rate and research on the urgency of deepening the reform of art education First-principles calculations of magnetic and mechanical properties of Fe-based nanocrystalline alloy Fe _{80}Si_{10}Nb_{6}B_{2}Cu_{2}The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Has the belt and road initiative boosted the resident consumption in cities along the domestic route? – evidence from credit card consumption MCM of Student’s Physical Health Based on Mathematical Cone Attitude control for the rigid spacecraft with the improved extended state observer Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Cognitive Computational Model Using Machine Learning Algorithm in Artificial Intelligence Environment Application of Higher-Order Ordinary Differential Equation Model in Financial Investment Stock Price Forecast Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Informatisation of educational reform based on fractional differential equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis Research on the control of quantitative economic management variables under the numerical method based on stochastic ordinary differential equations Network monitoring and processing accuracy of big data acquisition based on mathematical model of fractional differential equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression System dynamics model of output of ball mill Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Sensitivity Analysis of the Waterproof Performance of Elastic Rubber Gasket in Shield Tunnel Human gait modelling and tracking based on motion functionalisation Analysis and synthesis of function data of human movement The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Design of Morlet wavelet neural network to solve the non-linear influenza disease system Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Motion about equilibrium points in the Jupiter-Europa system with oblateness Fractional Differential Equations in Electronic Information Models Badminton players’ trajectory under numerical calculation method BIM Engineering Management Oriented to Curve Equation Model Optimal preview repetitive control for impulse-free continuous-time descriptor systems Development of main functional modules for MVB and its application in rail transit Study on the impact of forest fire prevention policy on the health of forest resources Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Value Creation of Real Estate Company Spin-off Property Service Company Listing Selection by differential mortality rates Digital model creation and image meticulous processing based on variational partial differential equation Dichotomy model based on the finite element differential equation in the educational informatisation teaching reform model Nonlinear Dissipative System Mathematical Equations in the Multi-regression Model of Information-based Teaching The modelling and implementation of the virtual 3D animation scene based on the geometric centre-of-mass algorithm The policy efficiency evaluation of the Beijing–Tianjin–Hebei regional government guidance fund based on the entropy method The transfer of stylised artistic images in eye movement experiments based on fuzzy differential equations Research on behavioural differences in the processing of tenant listing information: An eye-movement experiment A review of the treatment techniques of VOC Some classes of complete permutation polynomials in the form of ( x ^{pm}−x +δ )^{s}+ax ^{pm}+bx overF _{p2m}The consistency method of linguistic information and other four preference information in group decision-making Research on the willingness of Forest Land’s Management Rights transfer under the Beijing Forestry Development A mathematical model of the fractional differential method for structural design dynamics simulation of lower limb force movement step structure based on Sanda movement Fractal structure of magnetic island in tokamak plasma Numerical calculation and study of differential equations of muscle movement velocity based on martial articulation body ligament tension Study on the maximum value of flight distance based on the fractional differential equation for calculating the best path of shot put Sports intensity and energy consumption based on fractional linear regression equation Analysis of the properties of matrix rank and the relationship between matrix rank and matrix operations Study on Establishment and Improvement Strategy of Aviation Equipment Research on Financial Risk Early Warning of Listed Companies Based on Stochastic Effect Mode Characteristics of Mathematical Statistics Model of Student Emotion in College Physical Education Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Nonlinear strategic human resource management based on organisational mathematical model Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model