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Research on absolute ranging technology of resampling phase comparison method based on FMCW

  • Bin Sun EMAIL logo , Junfang Song , Chao Wang , Lanlan Li and Linqiu Tan
Published/Copyright: September 27, 2023
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Open Physics
From the journal Open Physics Volume 21 Issue 1

Abstract

As an advanced optical precision ranging method, FMCW (frequency-modulated continuous-wave) laser interferometric ranging technology can achieve a large-scale and high-precision absolute distance measurement, so it has high research and application value in the field of coordinate measurement in large space. How to improve the resolution and stability of laser interferometry is a research difficulty in this technology. Based on FMCW laser interferometry relative distance technology, a method for achieving absolute distance measurement through phase comparison after repeated sampling is proposed. Using the FMCW laser distance measuring system, two interference optical paths of measuring interferometer and auxiliary interferometer are constructed, respectively. Synchronous sampling is performed on two interference signals, and the phase ratio of the two interference signals is obtained through the extracted frequency information. The optical path difference of the unknown measurement interferometer is derived from the known auxiliary interferometer information. This method uses a triangular wave modulation laser to reduce the influence of Doppler frequency shift and eliminate quantization errors. By using the fixed-point repeated sampling, the optical path difference information is quickly extracted, and the nonlinear error of laser interference frequency modulation is effectively reduced.

Keywords: FMCW; absolute ranging; resampling; phase comparison method; laser interference

1 Introduction

Since the birth of laser, it has played an important role in the field of distance measurement with its unique multiple characteristics such as high collimation, high coherence, and modulability [1,2]. Frequency-modulated continuous-wave (FMCW) laser interferometric ranging is derived from light detection and ranging technology. This method has the advantages of large ranging range, high accuracy, searchability, and stability and reliability and has been widely applied in the fields of ultra-precision manufacturing and large-scale detection [3,4,5]. The basic principle of FMCW is to implement periodic continuous frequency modulation on the laser, and the modulated laser first loads the ranging information in the optical path system and then combines the beam to form interference. When the optical paths corresponding to the reference and signal optical paths of the interference system are not equal, there is a certain frequency difference between the combined two FMCW lasers, resulting in a dynamic beat frequency interference signal. By demodulating the initial phase change of the beat signal, the change of the measured light path difference can be obtained [6,7]. FMCW laser interferometry technology produces a dynamic beat interference signal, and it is easy to extract the phase subdivision, phase-moving direction resolution, and interference period count in the signal. Since the extracted phase can subdivide the wavelength of a laser, the FMCW laser interferometry technology can achieve high-precision distance measurement [8,9].

The measurement system uses a distributed feedback laser (DFB) semiconductor laser as the light source. The laser is equipped with an fiber grating filter, which ensures a stable output wavelength, extremely narrow linewidth, and excellent edge mode suppression ratio. Due to its small size and convenient installation, this device has been widely used. However, as a semiconductor device, it has defects such as poor temperature characteristics and easy noise generation [10,11,12]. In actual measurement, the laser is affected by factors such as modulation nonlinearity error, system calibration error, and Doppler frequency shift. These factors all greatly affect the ranging accuracy and stability of the FMCW ranging system, making it difficult to achieve stable and reliable engineering applications.

In order to solve the aforementioned problems, many scholars have put forward a lot of very effective methods, which mainly include early active compensation and late fitting treatment. Active compensation methods include pre-correction compensation and feedback frequency modulation technology and electro-optic phase-locked loop feedback correction technology [13,14,15]. Post-processing methods include signal frequency reference correction, optical frequency comb correction, and multi-optical path correction [16,17]. Naresh built an FMCW laser ranging optical path and designed a semiconductor laser source feedback control system with a ranging resolution of 1.5 mm [18]. Satyan et al. used a single-mode vertical cavity surface emitting laser to build a ranging system with a spatial ranging resolution of 250 μm and a range of 1 m [19]. Iiyama et al. improved the light source system using fiber optic self-mixing technology and measured targets at a distance of 1.5 m, achieving a ranging resolution of 31 μm [20]. Baumann used a frequency comb to correct the optical frequency of the laser, increasing the ranging resolution to 130 μm and the repetition accuracy to 6 nm [16]. The aforementioned method optimizes the signal extraction method and improves the measurement accuracy to a certain extent. However, the Doppler phase shift and nonlinear error caused by the large-scale laser frequency dynamic modulation are still a difficult problem that seriously affects the measurement accuracy and stability and has not been completely solved.

This manuscript proposes a method of fixed-point repeated sampling and phase comparison based on the correction technique of two optical paths to eliminate the nonlinear effects caused by laser modulation. The fixed-point repeated sampling algorithm is derived in detail. The correctness of the formula is verified by simulation analysis, and the corresponding auxiliary measurement optical path based on optical fiber is built by combining the algorithm. The experimental results show that the system can calculate the accuracy and reliability of the optical path difference and realize the high-precision and large-size measurement engineering application of FMCW laser interferometry absolute ranging.

2 FMCW ranging principle

As shown in Figure 1, the FMCW laser is collimated by a collimating lens and then divided by a coupler. The optical path where the fixed reflector is located is the reference beam, and the OPD (optical path difference) will not change. The optical path where the movable reflector is located is a signal beam. When the reflected beam from the movable reflector and the fixed reflector meets at the coupler, the two beams will interfere to generate a dynamic interference signal. The displacement of the movable reflector is calculated by the frequency and phase of the interference signal.

Figure 1 FMCW optical system diagram.
Figure 1

FMCW optical system diagram.

When these two waves interfere, the interference signal light intensity, the OPD between the reference wave and the signal wave in one modulation period, is as follows:

I ( OPD , t ) = I 0 [ 1 + V cos ( ω t + ϕ 0 ) ] ,

I ( OPD , t ) = I 0 1 + V cos 2 π Δ v v m OPD c t + 2 π λ 0 OPD

(1) = I 0 [ 1 + V cos ( 2 π v b t + φ b 0 ) ] ,

where Δ v is the optical frequency modulation excursion, v m is the frequency the modulation signal, c is the speed of light in free pace, λ 0 is the central optical wavelength, and v b and φ b 0 are the frequency and the initial phase of the beat signal, respectively.

Then, the absolute ranging based on frequency is as follows:

(2) ν b = Δ ν ν m OPD c ,

(3) L = OPD / 2 = 1 2 c Δ ν ν m ν b

3 Resampling phase comparison method for absolute ranging

In order to improve the accuracy of absolute distance measurement in large range and eliminate the quantization error caused by nonlinear frequency modulation and experimental calibration, a dual-optical path-measuring device based on a single-mode fiber is proposed. As shown in Figure 2, the basic structure of the dual optical path-ranging system is mainly composed of two parallel interference devices: one is the measuring interference device and the other is the auxiliary interference device. The DFB laser emits a triangular wave signal with the wavelength of 1,550 nm, which is divided into two parts according to the ratio of 1:1 by the coupler. The emitted light entering the measuring interference device is emitted through the circulator and collimating lens. The light reflected by the measuring target mirror and the light reflected by the half lens of the collimator form Michelson interference in the loop. The beat frequency signal formed by the interference is collected by photoelectric detector 1 through circulator 3. The laser entering the auxiliary interference device is divided into two parts by the coupler, enters the optical fiber with different optical path difference, and then combines the beams to form the Mach–Zehnder interferometer. The interference signal is collected by photodetector 2.

Figure 2 Schematic diagram of light path.
Figure 2

Schematic diagram of light path.

Analyzing the characteristics of measurement systems using frequency characteristics and decoupling important information hidden in physical phenomena is one of the important methods in signal processing technology [21–23]. In order to explain in detail, the working principle of resampling phase comparison absolute ranging, the acquisition method, is mathematically deduced. When the modulation frequency of the laser signal is v m and the frequency modulation range is Δ υ , the measuring interference device and the auxiliary interference device have the same light source.

Auxiliary interferometer distance:

(4) L r = OPD / 2 = 1 2 c Δ ν ν m ν br

Measuring interferometer distance:

(5) L m = OPD / 2 = 1 2 c Δ ν ν m ν bm

Then,

L m L r = ν bm ν br = ω m ω r ,

(6) L m = ν bm ν br L r

The absolute ranging calculation method based on beat frequency is adopted. Obviously, the measurement accuracy is restricted by frequency. Since the phase can subdivide the beat frequency signal frequency and improve the measurement accuracy, the absolute ranging calculation can be realized based on the sum of phases within the time window Δ t .

Auxiliary interferometer distance:

(7) L r = OPD 2 = 1 2 c 2 π Δ ν ν m Δ ϕ r Δ t = K Δ ϕ r Δ t

Measuring interferometer distance:

(8) L m = OPD 2 = 1 2 c 2 π Δ ν ν m Δ ϕ m Δ t = K Δ ϕ m Δ t .

Then, there is:

(9) L m L r = Δ ϕ m Δ ϕ r

(10) L m = Δ ϕ m Δ ϕ r L r

It can be seen from the aforementioned formula that if the optical path difference of the auxiliary interferometer is known in advance and the interference signals of the two interferometers are sampled synchronously, the optical path difference of the measuring interferometer can be obtained only by determining the ratio of the phase changes of the two interference signals in the same time period.

4 Experimental verification and analysis

In order to verify the correctness of the aforementioned methods, an absolute ranging experiment with a length of 2 m was designed in the standard vertebra laboratory environment. As shown in Figure 3, the distance between the precision electric drive guide rail and the collimating lens is 2 m on the Thorlabs optical platform. The measuring target mirror is installed on the electric driver, and the movable range is 0–1 m. The measuring beam is emitted from the collimator and reflected by the measuring target mirror and then combined with the half lens in the collimator to form the beat frequency signal of the measurement interferometer. Through channel 1 of the lower machine, the measurement signal is collected by the processor, and the moving distance of the target mirror is modulated and demodulated. It can be seen from the figure that the self-made auxiliary interferometer has a fixed cavity length, and the stable beat signal is collected and processed by channel 2 of the lower machine. The notebook computer is used as the host computer to collect two interferometer signals processed by the lower computer, and the results are statistically analyzed.

Figure 3 Measurement experiment.
Figure 3

Measurement experiment.

During the measurement, the modulation frequency of DFB laser is 100Hz, the amplitude of modulation voltage is 1.23 V, and the modulation current range is 39.9–83.1 mA. The average change rate of laser output light frequency with current is about 0.4 Ghz/mA, and the frequency modulation range (bandwidth) is about 18 GHz. During the experiment, the micro-displacement platform was controlled to drive the measurement target mirror to move at a uniform speed, and the position of the guide rail and the measurement results of the interferometer were recorded every 100 mm (1,000,000 μm). In the experiment, ten measurements were taken within the range of 1–2 m, using the position of the guide rail as a reference. The results recorded and obtained are shown in the measurement error curve of the laser interferometer as shown in Figure 6. It can be seen from Figure 4. that the error is ≤10 μm.

Figure 4 Linear error curve of interferometer.
Figure 4

Linear error curve of interferometer.

In a laboratory environment, adjust the range of the distance between the collimator and the measurement target mirror to be 1.3 m, and continuously collect 17,000 data in a static state. The stability scatter diagram is shown in Figure 5. Perform normal distribution processing on the scatter plot; as shown in Figure 6, within the 95% confidence interval, the maximum standard deviation is 2.656 μm.

Figure 5 Scatter diagram of interferometer stability.
Figure 5

Scatter diagram of interferometer stability.

Figure 6 Normal distribution.
Figure 6

Normal distribution.

It can be seen through experiments that this method can eliminate the influence of the quantization error substituted by the experimental calibration and realize the ranging resolution of 1 nm, and the measurement standard deviation exceeds 3 μm, which proves that the method has a high resolution and measurement repeatability. However, because the system has not been accurately calibrated, there are still systematic errors in the ranging results. Since the distance measurement experiment is implemented on a vibration isolation optical platform, external vibration and temperature changes have little effect on the experimental results. However, the system is more sensitive to vibration and temperature, and the impact of vibration and temperature must be considered in actual measurement, and vibration isolation design and temperature control are required.

5 Conclusion

FMCW laser interferometry ranging technology is to linearly modulate the frequency of the emitted laser, and the local oscillator signal interferes with the echo signal to form a beat frequency signal. The measurement distance is calculated by extracting the frequency information of the beat frequency signal. In the actual measurement, the laser has the problem of modulation nonlinearity, which causes the spectrum broadening of the measured signal seriously, which limits the ranging accuracy of the FMCW ranging system. In order to solve these problems, this study proposes an absolute ranging method based on the resampling phase comparison. Two signals of measuring interferometer and auxiliary interferometer are constructed by using dual optical path FMCW laser-ranging system. If two interferometers are sampled synchronously, the optical path difference can be obtained only by calculating the phase ratio of the two interferometers. The experimental results show that the nonlinear frequency modulation error of the laser is reduced by using fixed-point sampling because the quantization error caused by experimental calibration is eliminated, and this method effectively improves the accuracy and stability of measurement.

  1. Funding information: This work was supported by the National Natural Science Foundation of China (No. 62263028) and Shaanxi Natural Science Foundation of China (No. 2023-JC-YB-592/2023-JC-QN-0633).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The data used to support the findings of this study are available from the corresponding author upon request.

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Received: 2023-05-23
Revised: 2023-08-13
Accepted: 2023-08-28
Published Online: 2023-09-27

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2023-0112
Keywords for this article
FMCW; absolute ranging; resampling; phase comparison method; laser interference
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  1. Regular Articles
  2. Dynamic properties of the attachment oscillator arising in the nanophysics
  3. Parametric simulation of stagnation point flow of motile microorganism hybrid nanofluid across a circular cylinder with sinusoidal radius
  4. Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach
  5. Behaviour and onset of low-dimensional chaos with a periodically varying loss in single-mode homogeneously broadened laser
  6. Ammonia gas-sensing behavior of uniform nanostructured PPy film prepared by simple-straightforward in situ chemical vapor oxidation
  7. Analysis of the working mechanism and detection sensitivity of a flash detector
  8. Flat and bent branes with inner structure in two-field mimetic gravity
  9. Heat transfer analysis of the MHD stagnation-point flow of third-grade fluid over a porous sheet with thermal radiation effect: An algorithmic approach
  10. Weighted survival functional entropy and its properties
  11. Bioconvection effect in the Carreau nanofluid with Cattaneo–Christov heat flux using stagnation point flow in the entropy generation: Micromachines level study
  12. Study on the impulse mechanism of optical films formed by laser plasma shock waves
  13. Analysis of sweeping jet and film composite cooling using the decoupled model
  14. Research on the influence of trapezoidal magnetization of bonded magnetic ring on cogging torque
  15. Tripartite entanglement and entanglement transfer in a hybrid cavity magnomechanical system
  16. Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data
  17. Degradation of Vibrio cholerae from drinking water by the underwater capillary discharge
  18. Multiple Lie symmetry solutions for effects of viscous on magnetohydrodynamic flow and heat transfer in non-Newtonian thin film
  19. Thermal characterization of heat source (sink) on hybridized (Cu–Ag/EG) nanofluid flow via solid stretchable sheet
  20. Optimizing condition monitoring of ball bearings: An integrated approach using decision tree and extreme learning machine for effective decision-making
  21. Study on the inter-porosity transfer rate and producing degree of matrix in fractured-porous gas reservoirs
  22. Interstellar radiation as a Maxwell field: Improved numerical scheme and application to the spectral energy density
  23. Numerical study of hybridized Williamson nanofluid flow with TC4 and Nichrome over an extending surface
  24. Controlling the physical field using the shape function technique
  25. Significance of heat and mass transport in peristaltic flow of Jeffrey material subject to chemical reaction and radiation phenomenon through a tapered channel
  26. Complex dynamics of a sub-quadratic Lorenz-like system
  27. Stability control in a helicoidal spin–orbit-coupled open Bose–Bose mixture
  28. Research on WPD and DBSCAN-L-ISOMAP for circuit fault feature extraction
  29. Simulation for formation process of atomic orbitals by the finite difference time domain method based on the eight-element Dirac equation
  30. A modified power-law model: Properties, estimation, and applications
  31. Bayesian and non-Bayesian estimation of dynamic cumulative residual Tsallis entropy for moment exponential distribution under progressive censored type II
  32. Computational analysis and biomechanical study of Oldroyd-B fluid with homogeneous and heterogeneous reactions through a vertical non-uniform channel
  33. Predictability of machine learning framework in cross-section data
  34. Chaotic characteristics and mixing performance of pseudoplastic fluids in a stirred tank
  35. Isomorphic shut form valuation for quantum field theory and biological population models
  36. Vibration sensitivity minimization of an ultra-stable optical reference cavity based on orthogonal experimental design
  37. Effect of dysprosium on the radiation-shielding features of SiO2–PbO–B2O3 glasses
  38. Asymptotic formulations of anti-plane problems in pre-stressed compressible elastic laminates
  39. A study on soliton, lump solutions to a generalized (3+1)-dimensional Hirota--Satsuma--Ito equation
  40. Tangential electrostatic field at metal surfaces
  41. Bioconvective gyrotactic microorganisms in third-grade nanofluid flow over a Riga surface with stratification: An approach to entropy minimization
  42. Infrared spectroscopy for ageing assessment of insulating oils via dielectric loss factor and interfacial tension
  43. Influence of cationic surfactants on the growth of gypsum crystals
  44. Study on instability mechanism of KCl/PHPA drilling waste fluid
  45. Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media
  46. A novel compact highly sensitive non-invasive microwave antenna sensor for blood glucose monitoring
  47. Inspection of Couette and pressure-driven Poiseuille entropy-optimized dissipated flow in a suction/injection horizontal channel: Analytical solutions
  48. Conserved vectors and solutions of the two-dimensional potential KP equation
  49. The reciprocal linear effect, a new optical effect of the Sagnac type
  50. Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials
  51. The soliton solutions for stochastic Calogero–Bogoyavlenskii Schiff equation in plasma physics/fluid mechanics
  52. Research on absolute ranging technology of resampling phase comparison method based on FMCW
  53. Analysis of Cu and Zn contents in aluminum alloys by femtosecond laser-ablation spark-induced breakdown spectroscopy
  54. Nonsequential double ionization channels control of CO2 molecules with counter-rotating two-color circularly polarized laser field by laser wavelength
  55. Fractional-order modeling: Analysis of foam drainage and Fisher's equations
  56. Thermo-solutal Marangoni convective Darcy-Forchheimer bio-hybrid nanofluid flow over a permeable disk with activation energy: Analysis of interfacial nanolayer thickness
  57. Investigation on topology-optimized compressor piston by metal additive manufacturing technique: Analytical and numeric computational modeling using finite element analysis in ANSYS
  58. Breast cancer segmentation using a hybrid AttendSeg architecture combined with a gravitational clustering optimization algorithm using mathematical modelling
  59. On the localized and periodic solutions to the time-fractional Klein-Gordan equations: Optimal additive function method and new iterative method
  60. 3D thin-film nanofluid flow with heat transfer on an inclined disc by using HWCM
  61. Numerical study of static pressure on the sonochemistry characteristics of the gas bubble under acoustic excitation
  62. Optimal auxiliary function method for analyzing nonlinear system of coupled Schrödinger–KdV equation with Caputo operator
  63. Analysis of magnetized micropolar fluid subjected to generalized heat-mass transfer theories
  64. Does the Mott problem extend to Geiger counters?
  65. Stability analysis, phase plane analysis, and isolated soliton solution to the LGH equation in mathematical physics
  66. Effects of Joule heating and reaction mechanisms on couple stress fluid flow with peristalsis in the presence of a porous material through an inclined channel
  67. Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution
  68. Dynamical and physical characteristics of soliton solutions to the (2+1)-dimensional Konopelchenko–Dubrovsky system
  69. Study of fractional variable order COVID-19 environmental transformation model
  70. Sisko nanofluid flow through exponential stretching sheet with swimming of motile gyrotactic microorganisms: An application to nanoengineering
  71. Influence of the regularization scheme in the QCD phase diagram in the PNJL model
  72. Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior
  73. Computational analysis of reconstructing current and sag of three-phase overhead line based on the TMR sensor array
  74. Investigation of tripled sine-Gordon equation: Localized modes in multi-stacked long Josephson junctions
  75. High-sensitivity on-chip temperature sensor based on cascaded microring resonators
  76. Pathological study on uncertain numbers and proposed solutions for discrete fuzzy fractional order calculus
  77. Bifurcation, chaotic behavior, and traveling wave solution of stochastic coupled Konno–Oono equation with multiplicative noise in the Stratonovich sense
  78. Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity
  79. Numerical simulation and analysis of Airy's-type equation
  80. A homotopy perturbation method with Elzaki transformation for solving the fractional Biswas–Milovic model
  81. Heat transfer performance of magnetohydrodynamic multiphase nanofluid flow of Cu–Al2O3/H2O over a stretching cylinder
  82. ΛCDM and the principle of equivalence
  83. Axisymmetric stagnation-point flow of non-Newtonian nanomaterial and heat transport over a lubricated surface: Hybrid homotopy analysis method simulations
  84. HAM simulation for bioconvective magnetohydrodynamic flow of Walters-B fluid containing nanoparticles and microorganisms past a stretching sheet with velocity slip and convective conditions
  85. Coupled heat and mass transfer mathematical study for lubricated non-Newtonian nanomaterial conveying oblique stagnation point flow: A comparison of viscous and viscoelastic nanofluid model
  86. Power Topp–Leone exponential negative family of distributions with numerical illustrations to engineering and biological data
  87. Extracting solitary solutions of the nonlinear Kaup–Kupershmidt (KK) equation by analytical method
  88. A case study on the environmental and economic impact of photovoltaic systems in wastewater treatment plants
  89. Application of IoT network for marine wildlife surveillance
  90. Non-similar modeling and numerical simulations of microploar hybrid nanofluid adjacent to isothermal sphere
  91. Joint optimization of two-dimensional warranty period and maintenance strategy considering availability and cost constraints
  92. Numerical investigation of the flow characteristics involving dissipation and slip effects in a convectively nanofluid within a porous medium
  93. Spectral uncertainty analysis of grassland and its camouflage materials based on land-based hyperspectral images
  94. Application of low-altitude wind shear recognition algorithm and laser wind radar in aviation meteorological services
  95. Investigation of different structures of screw extruders on the flow in direct ink writing SiC slurry based on LBM
  96. Harmonic current suppression method of virtual DC motor based on fuzzy sliding mode
  97. Micropolar flow and heat transfer within a permeable channel using the successive linearization method
  98. Different lump k-soliton solutions to (2+1)-dimensional KdV system using Hirota binary Bell polynomials
  99. Investigation of nanomaterials in flow of non-Newtonian liquid toward a stretchable surface
  100. Weak beat frequency extraction method for photon Doppler signal with low signal-to-noise ratio
  101. Electrokinetic energy conversion of nanofluids in porous microtubes with Green’s function
  102. Examining the role of activation energy and convective boundary conditions in nanofluid behavior of Couette-Poiseuille flow
  103. Review Article
  104. Effects of stretching on phase transformation of PVDF and its copolymers: A review
  105. Special Issue on Transport phenomena and thermal analysis in micro/nano-scale structure surfaces - Part IV
  106. Prediction and monitoring model for farmland environmental system using soil sensor and neural network algorithm
  107. Special Issue on Advanced Topics on the Modelling and Assessment of Complicated Physical Phenomena - Part III
  108. Some standard and nonstandard finite difference schemes for a reaction–diffusion–chemotaxis model
  109. Special Issue on Advanced Energy Materials - Part II
  110. Rapid productivity prediction method for frac hits affected wells based on gas reservoir numerical simulation and probability method
  111. Special Issue on Novel Numerical and Analytical Techniques for Fractional Nonlinear Schrodinger Type - Part III
  112. Adomian decomposition method for solution of fourteenth order boundary value problems
  113. New soliton solutions of modified (3+1)-D Wazwaz–Benjamin–Bona–Mahony and (2+1)-D cubic Klein–Gordon equations using first integral method
  114. On traveling wave solutions to Manakov model with variable coefficients
  115. Rational approximation for solving Fredholm integro-differential equations by new algorithm
  116. Special Issue on Predicting pattern alterations in nature - Part I
  117. Modeling the monkeypox infection using the Mittag–Leffler kernel
  118. Spectral analysis of variable-order multi-terms fractional differential equations
  119. Special Issue on Nanomaterial utilization and structural optimization - Part I
  120. Heat treatment and tensile test of 3D-printed parts manufactured at different build orientations
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