Fractional nonlinear differential equations in chain solutions such as viscoelasticity, chaos and turbulence, aerospace, polymer materials, as well as non-Newtonian mechanics and other applied science fields, has a wide range of applications. The authors using the tools of nonlinear functional analysis, in order to study the existence theory and properties of solutions of several types of fractional nonlinear differential equations (groups), Misiats O et al., strive to find the conditions that make the corresponding nonlinear equation (group) solutions exist and are easier to verify or test, and the good properties of the solutions [1]; At the same time, we strive to construct an iterative approximation sequence that converges to the solution, and give an error estimation formula for the corresponding approximate solution. Fractional calculus has almost the same development history as integer calculus, it has a history of more than 300 years since it was proposed. In the past, it has been slow to develop because it has not been promoted by the practical application background, therefore, for a long time, only the research object of curiosity-driven mathematicians, without getting enough attention, until the early 1980s, after Tan Y et al. successfully used fractional derivatives to describe damping, people gradually realized that, fractional derivative is a powerful tool to describe viscoelasticity and other materials with “memory” and “genetic” characteristics. So far [2], the research on fractional calculus and its applications has gradually attracted the attention of many physical engineering researchers. After a period of development, Kanagawa T et al. excavated its many important applications, and gradually extended to such as diffusion and transport theory, chaos and turbulence, aerospace, viscoelasticity, chain solution of polymer materials, and non-Newtonian mechanics and other application fields [3]. According to classical Newtonian mechanics, space and time are continuous everywhere, therefore, integer-order differential operators can be appropriately used to describe basic physical quantities (such as velocity, momentum, force, etc.), therefore, the development and evolution process of many phenomena in nature, it is often described by integer order differential equations, such as heat conduction equation, wave equation, Hamilton equation, etc, people have made many brilliant achievements in many fields such as electromagnetism, classical mechanics, heat transfer and so on. However, unfortunately, the integer order differential operator has no global correlation, just is only a local concept, very inappropriate to describe natural phenomena with strong “memory” and “heredity”, therefore, using the classical integer order mathematical model, in order to describe and characterize those involved in memory and inheritance of processes, when the global correlation and dependency of the path are “anomalous”, it is often very limited, at this point, the fractional differential operator provides us with a good choice, it can just make up for the limitations and deficiencies of integer order operators in describing the above phenomena. Throughout the history of sound synthesis, from the electronic analog in the 1940s and 1950s, by the 80s and early 90s FM technology and wavetable synthesis technology, then to the physical models of the mid-1990s, all reflect the increasingly mature sound synthesis technology. As far as the current research situation is concerned, involvement in the field of sound synthesis technology is relatively shallow, and mainly focus on the research of speech acoustics and hydroacoustics, among them, the research on speech synthesis is relatively mature. Because the research on sound simulation of musical instruments requires cross-disciplinary talents, there are very few researches in this area. Compared with China, the development of musical instrument sound simulation in foreign countries is earlier, and it is relatively mature, and the instrument simulation technology of physical model is the main direction of current research.

Since the beginning of music synthesis research in 1940, there are several simulation synthesis technologies such as Wavetable Synthesis, Spectral Synthesis, Nonlinear Synthesis and Physical Model. Wavetable synthesis, which is based on samples extracted from sounds, with little regard for their physical properties. Spectral synthesis is to generate sound according to the time-frequency characteristics of the audio itself, the synthesized parameters are derived from the description of the desired waveform. Nonlinear synthesis is the use of complex synthesis algorithms to create rich sounds [4]. In contrast to spectral synthesis, the parameters of the nonlinear model have no direct relationship to the resulting waveform. The physical model describes the sound of the object according to the vibration equation of the object. Instead of simulating waveforms, it simulates the physical behavior of strings, drums, etc. Wavetable synthesis technology (WAVE TABLE) is one of the most widely used technologies in sound synthesis technology, it is mainly composed of two parts, one is the music event database, and the other is the sound playback. The specific implementation process is to sample the sounds emitted by various real musical instruments, store it in a digital music event database. When playing back, according to the needs of the synthesized music material, send instructions to the database, so as to find out the corresponding timbre information, then it is played back after being processed by the microprocessor or CPU. The playback toolset does more than just reproduce the original material, at the same time, it also performs pitch shifting, ooping, envelope and filtering, in order to achieve the effect of rich sound. The advantages of wavetable synthesis: First, the computational overhead is small; Second, it is easy to operate. The disadvantage is that a large data storage is required, and it is difficult to present different sounds due to the influence of pre-collected data.

The upper and lower solution method combined with the monotonic iteration technique is a powerful tool for analyzing nonlinear problems, its importance and compared with other analysis tools, its uniqueness and superiority, the advantage of this method is that, not only can prove the existence of the solution of the nonlinear problem under study, and can provide a monotonic iterative sequence and error estimation current that converges consistently to the solution, this tool has also been applied to solve the existence study of solutions to fractional nonlinear differential equations, recently, by applying a top-bottom solution method combined with a monotonic iterative technique, the initial value problem of nonlinear fractional differential equations.
^{3}, ^{3}, ^{α}^{β}^{γ}

The main purpose is to study the periodic boundary value problem

Obviously, for any ζ ∈ R, there is a unique solution _{ζ}^{−} (^{+} if

According to the physical sound mechanism of the piano, we can model the music simulation system, it is divided into two parts: excitation system and resonance system (as shown in Figure 1). The simulation implementation process, it is based on the superposition of a series of sine waves to simulate the vibration of strings, in order to make the sound simulation more realistic, the sound modification is carried out from two levels of time domain and frequency domain respectively. In the time domain, the envelope function obtained by fitting the time domain envelope extracted by the short-time RMS energy, in order to simulate the natural decay of the strings, so that the notes of the music can be connected harmoniously; In the frequency domain, the spectral envelope is the main manifestation of the resonant body, a filter bank for modeling the envelope of the musical audio spectrum extracted by the cepstral method, in order to achieve an effective simulation of the resonance system to synthesize musical sounds [7].

The basic sound source of the piano is the strings, and the player excites the strings to vibrate through the hammers, resulting in sound. We know that the vibration of the strings is superimposed in a sine wave mode, which envelopes the fundamental wave and the higher harmonic components, so the author will use the additive synthesis technique to simulate the vibration of the strings [8]. Next, we need to find the fundamental and harmonic frequency components, duration and other parameters required for each tone of the song to be played. By analogy, the frequency corresponding to each note in the musical notation is calculated as shown in Table 1 below.

Frequency table of each note

Note | 1 | 2 | 3 | 1 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

Fundamental frequency (Hz) | 261.62 | 293.65 | 329.62 | 261.62 | 261.62 | 293.65 | 329.62 |

Note | 1 | 3 | 4 | 5 | 3 | 4 | 5 |

Fundamental frequency (Hz) | 261.62 | 329.62 | 349.22 | 391 | 329.62 | 349.62 | 392 |

Secondly, according to the beat identification in Table 1, 4/4 represents a quarter note as one beat, and each section has four beats. In the first and second measures, each note occupies one beat, in the third measure, the first two notes each occupy one beat, and the third note occupies two beats. Knowing the number of beats per measure, we can determine the duration of each note in the music [9]. Assuming that the time of one beat is about 0.5s, the duration of each note in the music is shown in Table 2.

The duration of each note

Note | 1 | 2 | 3 | 1 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

Duration(s) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

Note | 1 | 3 | 4 | 5 | 3 | 4 | 5 |

Duration(s) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

Based only on the data of the fundamental frequency and duration of each note in the music obtained above, not enough to model string vibrations, because there is no overtone component added in this model, the simulated music will lack a certain sense of heaviness, it is very different from the sound of the real instrument. Therefore, to add the corresponding harmonic components to the simulated model, the harmonic frequencies of the musical sounds are almost integer multiples of the fundamental frequency, and there is a certain relationship between their amplitudes, harmonic components can be added directly using the fitted curve function. Some researchers have used the weighted Cauchy function, simulates the 5-octave resonance of a piano, the simulation of timbre was realized, and later professionals found that its overtone was insufficient, and its timbre expression was not enough [10]. Then, in the paper on the design and implementation of the piano timbre recognition and electronic synthesis system, the timbre feature matrix extraction method is used, under the condition of considering the balance of effect and efficiency, the musical simulation of the 25th octave harmonic in the timbre is realized, the sound reproduction is impeccable, making up for the lack of sound of the 5th octave of the Cauchy function. But for a piano with rich overtones, it is impossible for the number and amplitude of overtones corresponding to each tone to be the same, in Cheng Meifang's research experiment, the timbre feature matrix of a tone was extracted, tone emulation in place of the characteristics of all the tones in the song, obviously not convincing, can not show the unique spectral characteristics of each tone in the composition. Therefore, in order to grasp the spectral characteristics of each tones in the music, we collected the single-tone materials of the piano C major (do, re, mi, fa, so, la, si), fourier transform is performed, respectively, obtain the frequency and normalized amplitude of each harmonic. For the above phenomenon, we know that it is caused by the inability to smoothly transition between adjacent notes of the tones. Therefore, on the basis of the above experiments, we introduce the exponential envelope and polyline envelope functions respectively, and the function of the newly designed segmented envelope is modified in the time domain, and make a simple loop process acting on each note in the song, the length and amplitude of each envelope curve, respectively, determined by the duration and amplitude of the individual notes.

Wherein (a) is the musical tone simulated by the exponential envelope; (b) is the musical tone simulated by the polygonal envelope; (c) Tones simulated for the newly designed segmental envelope curve; (d) Simultaneously modifies the simulated tones for the segmented envelope curve and filter. The following will give a detailed analysis of each simulation effect:

It can be heard from the playing music, exponential decay technology can well eliminate the transition noise between each note, decay speed determined by playing time, makes the music sound more ups and downs, lively and powerful. But through audition judgment, it is found that there is still a sudden change in phase between adjacent notes, the resulting slight noise, the transition between the two notes is not very smooth.

The application of polyline envelope technology, it is guaranteed that when the amplitude of the vibration of the previous note is attenuated to 0, only start the vibration of the next note, which can effectively eliminate the noise between note transitions. But in the process of playing the music, careful professional musicians found that, although the polyline envelope method can make the transition between the notes of the music smooth, but the simulated music sounds more blunt, it cannot show the softness of piano music, and the amplitude change does not conform to the phenomenon of stringed instruments.

From the action diagram of the newly designed piecewise envelope curve, the envelope function curve changes softly, and the attack stage is very short in time, it can increase the vibration amplitude to the maximum, ensuring the efficient pronunciation of the piano, as time goes on, the amplitude gradually decays, and when the sound of the note ends, the amplitude decays to 0, in order to prepare for the pronunciation of the next note, it is better to avoid the noise generated by the transition between adjacent notes. Discovered by audition, the musical sound effect simulated by the piecewise envelope function, significantly better than exponential envelopes and polyline envelopes, the algorithm not only makes the transition between notes smooth, but also softens the change of each tone itself, completely eliminating the noise, at the same time, it also shows that the piano sound is pronounced from the strings being struck, the whole process of soft change to the disappearance of sound intensity.

From the newly designed piecewise envelope function, it can be seen from the effect diagram that the filter modeled by the spectral envelope is modified at the same time, this method not only makes the connection and harmony among the musical notes, but also enhances the sound intensity of some signals. The music sounds undulating and orderly, very close to a real piano.

In order to verify the effectiveness of the algorithm, this subsection further evaluates. One is to judge the readiness rate by detecting the pitch frequency of the musical tone; The second is to compare the difference between the formant frequency of the simulated music and the formant peak of the real piano music; The third is to compare the effect achieved by the analysis of the spectrogram of the simulated musical tone signal in each link.

First, the linear residual cepstrum method is used, perform fundamental frequency detection on the musical tone signal of the piano simulation, this algorithm is a research hotspot in the field of audio processing, which can well avoid the influence of channel features and noise, make the test results more prepared. The test results are listed in Table 3, and compared with the theoretical value of the pitch frequency of each note in the notation chart, get its accuracy. For simplicity, only some of the pitch frequencies of the notes are shown in the Table.

Test results

Each note in the notation | Theoretical value (HZ) | Detection value (HZ) | Accuracy(%) |
---|---|---|---|

1 | 216.62 | 262.12 | 99.8 |

2 | 293.65 | 293.01 | 99.7 |

3 | 329.62 | 330.13 | 99.8 |

1 | 261.62 | 262.12 | 99.8 |

3 | 329.62 | 331.56 | 99.4 |

4 | 349.22 | 352.7 | 98.9 |

5 | 390 | 393.74 | 99.5 |

From the data detected in the Table, it can be seen that, the musical sound signal simulated by the author's method, and the theoretical value of each note of the real piano instrument collected, very close, with an accuracy rate of more than 98%, this data proves the accuracy of the simulated piano sound.

Based on the analysis of the vibration and attenuation characteristics of piano strings, and on the theoretical basis of the discussion on the resonance effect of the resonance system, the model of the piano music simulation system is constructed, a class with multi-order fractional derivatives is studied, and the perturbation term of the fractional integral boundary value problem of linear impulse differential equations, and one by one for the excitation system and resonance system in the model, carry out simulation modeling and experimental verification. It is found that there is a transition noise between adjacent notes, in order to eliminate this noise, try the exponential envelope technique, the polyline envelope technique separately, improve it, then take the multi-order fractional derivative to be studied, and the solution of the nonlinear impulse differential equation with the perturbation term, transformed into the fixed point problem of the operator, then by using Schauder's fixed point theorem and Banach's compression mapping principle, given the nonlinear multifractional impulse differential equation, the conditions for the existence of at least one solution and a unique solution, although these methods can eliminate the switching noise between notes, however, this method will make the simulated music sound blunt, which is not in line with the characteristics of piano music. Therefore, according to the time-domain amplitude envelope extracted from the collected piano monophonic signal, a piecewise envelope function is obtained, this envelope was found to be the perfect interpretation of the piano's entire process from articulation to decay. Secondly, the author also combined the formant point of the spectral envelope, which reflects the important information of the piano resonance system, and the sound effect EQ to design a filter, modify the simulated musical tone signal in the frequency domain, improves the timbre of musical tones, making them sound more harmonious, close to the real instrument. Finally, from the pitch frequency detection, there are three aspects of formant frequency estimation and spectrogram analysis, the author's algorithm is subjected to subjective evaluation experiments, and the experimental results verify the effectiveness of musical sound simulation.

#### Frequency table of each note

Note | 1 | 2 | 3 | 1 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

Fundamental frequency (Hz) | 261.62 | 293.65 | 329.62 | 261.62 | 261.62 | 293.65 | 329.62 |

Note | 1 | 3 | 4 | 5 | 3 | 4 | 5 |

Fundamental frequency (Hz) | 261.62 | 329.62 | 349.22 | 391 | 329.62 | 349.62 | 392 |

#### Test results

Each note in the notation | Theoretical value (HZ) | Detection value (HZ) | Accuracy(%) |
---|---|---|---|

1 | 216.62 | 262.12 | 99.8 |

2 | 293.65 | 293.01 | 99.7 |

3 | 329.62 | 330.13 | 99.8 |

1 | 261.62 | 262.12 | 99.8 |

3 | 329.62 | 331.56 | 99.4 |

4 | 349.22 | 352.7 | 98.9 |

5 | 390 | 393.74 | 99.5 |

#### The duration of each note

Note | 1 | 2 | 3 | 1 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

Duration(s) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

Note | 1 | 3 | 4 | 5 | 3 | 4 | 5 |

Duration(s) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

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Optimization and Mathematical Modeling for English-Chinese Cross-Language Information Retrieval 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 research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontal-torsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation MCM of Student’s Physical Health Based on Mathematical Cone 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 Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing 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 Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference 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 Animation VR scene mosaic modeling based on generalized Laplacian equation 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 The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation 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 Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation 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 Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors 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 Human gait modelling and tracking based on motion functionalisation 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 Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability 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Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp -groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edge-based finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leader-follower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSR-AHP model A study of local smoothness-informed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H _{2}-N_{2}atmosphereAlgorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion 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 Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model