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In the spectral analysis of operators associated with Sturm–Liouville-type boundary value problems for fractional differential equations, the problem of positive definiteness or the problem of Hermitian nonnegativity of the corresponding kernels plays an important role. The present paper is mainly devoted to this problem. It should be noted that the operators under study are non-self-adjoint, their spectral structure is not well investigated. In this paper we use various methods to prove the Hermitian non-negativity of the studied kernels; in particular, a study of matrices that approximate the Green's function of the boundary value problem for a differential equation of fractional order is carried out. Using the well-known Livshits theorem, it is shown that the system of eigenfunctions of considered operator is complete in the space L 2 (0 , 1) . Generally speaking, it should be noted that this very important problem turned out to be very difficult. [ABSTRACT FROM AUTHOR]

Every application of integration by parts can be done with a tabular method. The trick is to identify and consider each new integral in the table before deciding how to proceed. This paper supplements a classic introduction to integration by parts with a particular tabular method called Row Integration by Parts (RIP). Approaches to tabular methods found online and in the literature often fail to clearly identify new integrals since the focus is on constructing columns, thus new integrals are ignored. With RIP, each integral is identified with a row and tables are built one row at a time. This allows RIP to be an efficient bookkeeping method and to apply to a much wider range of functions than those typically considered with other tabular methods. [ABSTRACT FROM AUTHOR]

An active inerter-based suspension with acceleration feedback control is proposed in this paper, the time delay generated in the controllers and actuators is considered, which constitutes the time-delayed active inerter-based (TDA-IB) suspension. The dynamic equation of the TDA-IB suspension is established and is a neutral type of delay differential equation (NDDE) in which the time delay exists in the highestorder derivative. The stability analysis is conducted by calculating the number of unstable characteristic roots based on the definite integral stability method, the stable and unstable regions are determined. The effect of time delay and feedback gain on the dynamic performance of the TDA-IB suspension under harmonic, random, and shock excitations is studied in detail and compared with the parallel-connected inerterbased (PC-IB) and traditional suspensions. The results show that the TDA-IB suspension is asymptotically stable for smaller feedback gain and time delay, through increasing the feedback gain, the stable regions shrink, and a smaller time delay could cause the system to become unstable. Furthermore, the time delay could regulate the resonance peak around the unsprung mass natural frequency and generate multiple high-frequency resonance peaks. If the time delay is chosen appropriately and falls into the stable range, the TDA-IB suspension could improve the dynamic performance for the suspension stroke and dynamic tire load while having a deterioration for the vehicle body acceleration compared with the PC-IB and traditional suspensions. [ABSTRACT FROM AUTHOR]

In this article new estimates on some quadrature rules are given using weighted Hermite-Hadamard inequality for higher order convex functions and weighted version of the integral identity expressed by w-harmonic sequences of functions. Obtained results are applied to weighted one-point formula for numerical integration in order to derive new estimates of the definite integral values. [ABSTRACT FROM AUTHOR]

Introduction/purpose: An approximate approach to definite integral calculation has been an attractive problem continuously since the creation of integration due to practical needs in scientific and engineering areas. In most practical cases, the integrand is complex, which leads to a difficulty of obtaining an exact value of integration, so an approximate value of the definite integral with certain accuracy is satisfactory for practical applications. In this paper, an efficient approach for calculating a definite integral with a small number of sampling points is proposed based on the uniform design method from the viewpoint of practical application. Methods: The distribution of sampling points in its single peak domain is deterministic and uniform, which follows the rule of the uniform design method and good lattice points. [ABSTRACT FROM AUTHOR]

In this article we showed how the following improper real integral I n = ∫ 0 ∞ d x x n + 1 = π / n sin π / n which is typically evaluated using complex variables and contour integration for general n can actually be evaluated using the integration by partial fractions technique of single variable Calculus. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study the definite neutrosophic integrals, where the neutrosophic integrals are defined, in addition, set of theories and properties related to them were discussed, also, applications of the definite neutrosophic integrals were introduced, such as area of neutrosophic curves, length of neutrosophic curve and volumes of neutrosophic revolution. Where detailed examples were given to clarify each case. [ABSTRACT FROM AUTHOR]

Probabilistic hesitant fuzzy sets (PHFSs) add the probability value corresponding to each degree of membership on the basis of hesitant fuzzy sets, so as to express the initial decision information given by experts more accurately and comprehensively. In this article, we mainly study how to integrate large-scare probabilistic hesitant fuzzy information more efficiently. We first discuss some basic operation laws of probabilistic hesitant fuzzy numbers, based on which the concepts of continuous PHFSs and continuous probabilistic hesitant fuzzy functions (c-PHFFs) are defined. They are the main objects of our research. We further explore definite integrals of the c-PHFFs and their related properties. They have direct and powerful applications in continuous probabilistic hesitant fuzzy environments, and lay the foundation for subsequent theoretical analysis. Based on the weight density function, we finally get the weighted-integral operator of continuous probabilistic hesitant fuzzy information. Then, some important properties of this integration operator are studied, including normalization, monotonicity, boundedness, etc. We are also devoted to revealing the inner connection between the continuous probabilistic hesitant fuzzy weighted-integral operator and the probabilistic hesitant fuzzy weighted averaging operator, the latter is usually used when dealing with discrete information. At last, we state why it is necessary to introduce a novel aggregation method based on continuous probabilistic hesitant fuzzy definite integrals, and in turn provide an application of the proposed method to prove its validity and rationality. [ABSTRACT FROM AUTHOR]

This work aims to investigate the dependence of gravitational lensing properties on the lens redshift and source redshift. The angular diameter distance hereafter referred to as ADD has been determined using two different numerical integral methods, Simpson's rule, and definite integral methods. Both of those two methods gave identical results. In addition, observational data of gravitational Lensing systems have been used to find the most probable value of lens redshift and source redshift. The result showed that the lens redshift and source redshift are more likely to occur in the ranges of zL=0.2-0.6 and zS=1-3, respectively. Einstein radius and the critical surface mass density have been studied for those ranges of lens and source redshift. The result clearly showed that the redshifts and distances affect the lensing properties. This indicates that observations of gravitational lensing can be efficiently used to probe the distances and redshifts [ABSTRACT FROM AUTHOR]

The method of brackets, developed in the context of evaluation of integrals coming from Feynman diagrams, is a procedure to evaluate definite integrals over the half-line. This method consists of a small number of operational rules devoted to convert the integral into a bracket series. A second small set of rules evaluates this bracket series and produces the result as a regular series. The work presented here combines this method with the classical Mellin transform to extend the class of integrands where the method of brackets can be applied. A selected number of examples are used to illustrate this procedure.

In the article, the authors discuss the mathematical method of integration that can be used in the calculation of Riemann integrals as proposed by Joseph Liouville (1809-1882). Topics include how Liouville discovered the classes of integrals that could be transformed or reduced to other integrals and the concept developed by P. L. Chebyshev of systematically classifying the procedures of integration of some classes of functions.

this manuscript, the authors derived a definite integral involving the logarithmic function, function of powers and polynomials in terms of the Lerch function. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers. [ABSTRACT FROM AUTHOR]

In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals. [ABSTRACT FROM AUTHOR]

Closed expressions using the Lerch function for a definite integral are derived and evaluated. Some of these closed expressions are given in Gradshteyn and Ryzhik. Some special cases of the integral are derived and discussed. The majority of the results in this work are new. [ABSTRACT FROM AUTHOR]

In this paper, non-classical numerical schemes are proposed for the approximation of Cauchy type oscillatory and strongly singular integrals in complex plane. The schemes are developed by incorporating classical quadrature rule meant for the Cauchy type complex singular integrals over a line segment in complex plane with a quasi exact quadrature method meant for the numerical integration of complex definite integrals with an oscillatory weight function. The error bounds are established and the schemes are numerically validated using a set of standard test integrals.Numerical results show that these schemes are efficient. [ABSTRACT FROM AUTHOR]

The stationary states of the quantum harmonic oscillator are properly determined by means of the unilateral Fourier transform without having to recourse to the properties of the confluent hypergeometric functions. This simpler procedure is reminiscent of the algebraic method based on the ladder operators and depends on the value of just one tabulated definite integral related to the ground state. [ABSTRACT FROM AUTHOR]

This article proposes a rotor position estimation method for brushless electrically excited synchronous starter/generator (BEESSG) at low speed region. Based on the multistage characteristics of BEESSG, two high frequency pulsating voltages with different frequencies are injected into phase $\alpha$ and phase $\beta$ of stator of the main machine simultaneously. The rotor position is estimated by the high frequency induced signals extracted from the stator windings of the main exciter through a position estimator. Low-pass filters are replaced by an auxiliary square wave sampling method to demodulate the rotor position from high frequency induced signals. Due to the amplitude unbalance of demodulated position related signals. A novel first quadrant definite integral method is proposed in this article for position compensation. Furthermore, the effectiveness and feasibility of the proposed method are verified by experimental results. [ABSTRACT FROM AUTHOR]

Because of the low surface roughness of three-dimensional (3D) printed microchannels, in some environments, the use requirements cannot be met, and, at the same time, they are difficult to process. Therefore, a surface roughness analysis of 3D printed microchannels and post-processing using abrasive flow technology are proposed. The value of the inner surface roughness of the straight pipe part was calculated by using the least-squares method combined with the definite integral. Using the equal area principle, MATLAB curve fitting was used to numerically calculate the semicircular pipe section, and the relationship between the value of surface roughness and the bending radius of the scanning layer thickness is given. Use MATLAB image processing technology to study the processing area. The roughness of the inner wall of the pipeline was analyzed by laser confocal, stereo, and scanning electron microscopes. The results show that the roughness of the inner surface of the pipe increases with an increase in the thickness of the sweeping layer, an increase in the inclination angle, and a decrease in the radius of curvature. In the case of abrasive flow processing, the longer the processing time, the greater the grinding amount and the greater the amount of grinding outside the pipe wall in each one-way machining; further, the processing has obvious directionality. [ABSTRACT FROM AUTHOR]

In the paper, by virtue of the convolution theorem for the Laplace transforms, with the aid of three monotonicity rules for the ratios of two functions, of two definite integrals, and of two Laplace transforms, in terms of the majorization, and in the light of other analytic techniques, the author presents decreasing properties of two ratios defined by three and four polygamma functions. [ABSTRACT FROM AUTHOR]

The method of brackets is a method for the evaluation of definite integrals based on a small number of rules. This is employed here for the evaluation of Mellin–Barnes integral. The fundamental idea is to transform these integral representations into a bracket series to obtain their values. The expansion of the gamma function in such a series constitute the main part of this new application. The power and flexibility of this procedure is illustrated with a variety of examples. [ABSTRACT FROM AUTHOR]

Definite integral equation solvers were hitherto developed using software which take long time to generate results. The complexity of designs and verifications, due to advancement in calculus and linear algebra, data dependency in designs, and the need for parallelism in models, necessitated the need for faster, accurate, and reliable hardware-based solvers. This will reduce latency and excessive memory requirement in generating values of definite integrals. The Field Programmable Gate Array (FPGA)-Based Definite Integral Solver developed in this research is able to solve definite integral equations with less resource utilization (1.08% look up tables (LUT), 0.07% flip flops (FF), 10.81% digital signal processing (DSP) and 23.86% input-output (IO)) and hence minimum switching activities at lower power (Signals: 0.421W, Logic: 0.345W, Digital Signal Processing: 1.904W and Input/Output: 0.015W) that translates to fast and accurate results generation. The power and resource utilization evaluation on the solver and the performance comparison with similar works showed that excessive memory consumption and resource utilization has been reduced significantly by the design. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the problem of approximation of a definite integral over a ball of a given function f in d -dimensional space when, rather than function evaluations, a number of integrals over certain (d − 1) -dimensional hyperspheres are only available. In this context several families of 'extended' multidimensional integration formulas based on a weighted sum of integrals over some hyperspheres can be defined. The special cases include multivariate analogues of the well-known midpoint rule and the trapezoidal rule. Basic properties of these families are derived, in particular, we show that they all satisfy a multivariate version of Hermite–Hadamard inequality. As an immediate consequence of this inequality, we derive explicit expressions of the best constants, which appear in their optimal error estimates. Theoretical and numerical results show that the proposed method reaches at least the second order of approximation. We present several numerical examples to illustrate various features of these new cubature formulas. [ABSTRACT FROM AUTHOR]

An investigation into a family of definite integrals contain- ing log-polylog functions will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and may be represented as a BBP-type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function. [ABSTRACT FROM AUTHOR]

The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application. A very simple formulation combines the derivation operators for functions and for expressions in a manner not found elsewhere due to common confusion between them. Some issues are rooted more deeply than others and are discussed in a broader perspective, starting with the function concept. The approach is illustrated using various applications including the transport equation, partial derivatives of a definite integral, and the distortionless (but not lossless) transmission line. This note is suitable for a lecture in any first-year course covering partial derivatives, as a complement to the other course material. [ABSTRACT FROM AUTHOR]

This article is concerned with establishing some new linearization formulas of the modified Jacobi polynomials of certain parameters. We prove that the linearization coefficients involve hypergeometric functions of the type 4F3(1). Moreover, we show that the linearization coefficients can be reduced in several cases by either utilizing certain standard formulas, and in particular Pfaff-Saalschütz identity and Watson's theorem, or via employing the symbolic algebraic algorithms of Zeilberger, Petkovsek, and van Hoeij. New formulas for some definite integrals are obtained with the aid of the developed linearization formulas. [ABSTRACT FROM AUTHOR]

In this paper, the conception of centroid of n-dimensional fuzzy number is introduced via regarding its membership function as the density function on its support set, and some properties of it are obtained. Compared with the mean of the multi dimensional fuzzy number, the centroid takes into account the overall relationship between the edge membership functions of the membership function of the multi dimensional fuzzy number. Therefore, it can approximate (characterize) the fuzzy number more objectively and reasonably than using the mean of the multi dimensional fuzzy number. The most important work of this paper is that for two special kinds of multi dimensional fuzzy numbers (fuzzy n-cell numbers and fuzzy n-ellipsoid numbers), we respectively give calculation formulas, which can be used conveniently in application since the formulas are based on a definite integral of the level set functions of the multi dimensional fuzzy number on the unit interval [0,1], rather than the multiple integral of the membership function of the multi dimensional fuzzy number itself on its support set. Then, by using the calculation formulas, we obtain another special property of the centroid for fuzzy n-cell number and fuzzy n-ellipsoid number. Finally, as an example of application, by using the centroid of multi dimensional fuzzy number, we define a fuzzy order on n-dimensional fuzzy number space, which can be used to rank uncertain or imprecise multichannel digital information. [ABSTRACT FROM AUTHOR]

Energy consumption plays a key role in economic development for all countries. Keeping up with the future trend of energy consumption is essential for governments and energy companies. In this research, the primary energy consumption for Saudi Arabia, India, Philippines, and Vietnam is systematically studied using different forecasting models. Based on the actual data derived from the year 2006 to 2016, a novel discrete grey forecasting model termed NDGMS(1,1,k,c) is proposed where the Simpson numerical integration formula is applied to construct the background value. The expressions of the present model are all derived and then, its unbiased property is proved. As demonstrated by the results, the NDGMS(1,1,k,c) model can achieve better prediction accuracy than other forecasting models, and it is quite applicable to predicting a sequence based on homogeneous/non-homogeneous exponential law. [ABSTRACT FROM AUTHOR]

This note deals with some variants of the integral mean value theorem. Mainly a variant of Sahoo's theorem and a variant of Wayment's theorem were proved. Our approach is rather elementary and does not use advanced techniques from analysis. The simple auxiliary functions were used to prove the results. [ABSTRACT FROM AUTHOR]

A definite integral involving the power square root of an algebraic function is derived in terms of the Lerch function. A table consisting of mostly new results is given for easy reading. The majority of the results in this work are new. [ABSTRACT FROM AUTHOR]

This current paper is a table of definite integrals involving hyperbolic and logarithmic functions expressed in terms of the incomplete gamma function and fundamental constants. All the results in this work are new. [ABSTRACT FROM AUTHOR]

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan’s constant and π. [ABSTRACT FROM AUTHOR]

In this paper, a series of new inequalities for definite integral are constructed. By rigorous analysis, the sufficient and necessary conditions for which the equality holds are carried out. Additionally, the optimal constant factors for these new inequalities are obtained. [ABSTRACT FROM AUTHOR]

Monte Carlo integration is a technique for numerically estimating a definite integral by stochastically sampling its integrand. These samples can be averaged to make an improved estimate, and the progressive estimates form a sequence that converges to the integral value on the limit. Unfortunately, the sequence of Monte Carlo estimates converges at a rate of O(), where n denotes the sample count, effectively slowing down as more samples are drawn. To overcome this, we can apply sequence transformation, which transforms one converging sequence into another with the goal of accelerating the rate of convergence. However, analytically finding such a transformation for Monte Carlo estimates can be challenging, due to both the stochastic nature of the sequence, and the complexity of the integrand. In this paper, we propose to leverage neural networks to learn sequence transformations that improve the convergence of the progressive estimates of Monte Carlo integration. We demonstrate the effectiveness of our method on several canonical 1D integration problems as well as applications in light transport simulation. [ABSTRACT FROM AUTHOR]

We give three elementary proofs of a nice equality of definite integrals, recently proven by Ekhad, Zeilberger, and Zudilin. The equality arises in the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a generalization together with an equally elementary proof and discuss some consequences. [ABSTRACT FROM AUTHOR]

As an economical and effective testing method, real-time hybrid simulation (RTHS) can be understood as a dynamic system with substructure feedback, where the time delay in feedback caused by servo-hydraulics may destabilize the system. Existing research does not realize that the time delay in each component has a different effect on stability. This study categorizes RTHS into neutral or retarded delay systems to differentiate their pole distribution and explore how time delay affects the stability. The analysis results show that (1) time delay introduces infinite pairs of conjugate characteristic roots into both neutral and retarded systems but results in different patterns of root distribution; (2) the characteristic roots approach based on the rational function approximation is only suitable for stability analysis of a retarded delay system, not for a neutral delay system; and (3) the definite integral stability analysis approach can directly determine the stability of both retarded and neutral delay systems without solving the transcendental equation. The computational simulation demonstrated the difference between retarded and neutral delay systems and the effectiveness of the new approach. This study provides fresh insights for the dynamic analysis of RTHS system with time delay. [ABSTRACT FROM AUTHOR]

In this article, we have investigated certain definite integrals and various integral transforms of the generalized multi-index Bessel function, such as Euler transform, Laplace transform, Whittaker transform, K-transform and Fourier transforms. Also found the applications of the problem on fractional kinetic equation pertaining to the generalized multi-index Bessel function using the Sumudu transform technique. Mittage-Leffler function is used to express the results of the solutions of fractional kinetic equation as well as its special cases. The results obtained are significance in applied problems of science, engineering and technology. [ABSTRACT FROM AUTHOR]

The importance of calculus in mathematics, mathematics education, and other disciplines and the necessity of developing students' conceptual understanding regarding integral, which is one of the major concepts in calculus course, are among the issues emphasized by researchers. Thus, the purposes of this study were to examine how pre-service middle school mathematics teachers describe definite integral and indefinite integral and also to what extent they can see the relation between definite integral and indefinite integral. For these purposes, 173 pre-service middle school mathematics teachers were asked to answer three questions. According to the findings, the concepts pre-service middle school mathematics teachers mentioned while describing both definite and indefinite integral are similar which are bound, notation, mathematical formula, example, area, volume, antiderivative, calculation process, the form of result (number, function, algebraic expression or unknown), and the constant c. It was also seen that mostly mentioned concept is bound for each type of integral. However, the minority of them presented evidence regarding the relation between definite integral and indefinite integral in their responses. [ABSTRACT FROM AUTHOR]

In this paper, we provide criteria to determine whether a given integral quadratic lattice over a ring Z p of p -adic integers is Z p -universal or primitively Z p -universal. The criteria are stated explicitly in terms of a Jordan splitting of the lattice. As an application of the local criteria, we complete the determination of the universal positive definite classically integral quaternary quadratic forms that are almost primitively universal. [ABSTRACT FROM AUTHOR]

In this paper, we obtain analytical solutions of an unsolved integral 𝐑 S ⁢ (m , n) {\mathbf{R}_{S}(m,n)} of Srinivasa Ramanujan [S. Ramanujan, Some definite integrals connected with Gauss's sums, Mess. Math. 44 1915, 75–86] with suitable convergence conditions in terms of Meijer's G-function of one variable, by using Mellin–Barnes type contour integral representations of the sine function, Laplace transform method and some algebraic properties of Pochhammer's symbol. Also, we have given some generalizations of Ramanujan's integral 𝐑 S ⁢ (m , n) {\mathbf{R}_{S}(m,n)} in the form of integrals ℧ S * ⁢ (υ , b , c , λ , y) {\mho_{S}^{*}(\upsilon,b,c,\lambda,y)} , Ξ S ⁢ (υ , b , c , λ , y) {\Xi_{S}(\upsilon,b,c,\lambda,y)} , ∇ S ⁡ (υ , b , c , λ , y) {\nabla_{S}(\upsilon,b,c,\lambda,y)} and ℧ S ⁢ (υ , b , λ , y) {\mho_{S}(\upsilon,b,\lambda,y)} with suitable convergence conditions and solved them in terms of Meijer's G-functions. Moreover, as applications of Ramanujan's integral 𝐑 S ⁢ (m , n) {\mathbf{R}_{S}(m,n)} , the three new infinite summation formulas associated with Meijer's G-function are obtained. [ABSTRACT FROM AUTHOR]

In the study of the durability of concrete structures, the chloride ion diffusion coefficient (D t) is a key parameter that directly affects the prediction of concrete's service life. This study aims to explore the relationship between the chloride ion diffusion coefficient in concrete and the "curing age (t ref)" and "exposure time (t)." By conducting long-term curing and seawater exposure experiments on three different mixes of high-performance concrete samples, we collected experimental data with a curing age of 3 years and a seawater exposure time of 4.5 years. Through in-depth analysis of these data, we found that there is a significant power function relationship between the chloride ion diffusion coefficient and exposure time, while the relationship with curing age is not significant. This finding suggests that in the derivation of the concrete chloride ion diffusion theory model, the time range of the definite integral of D t should be from 0 to t, rather than from t ref to t ref +t. Furthermore, we have validated the concrete life prediction model using actual data from the Swedish Marine Exposure Station, further confirming the accuracy and reliability of our research results. This study provides a more scientific and accurate theoretical basis for the durability assessment and service life prediction of concrete structures. • ClinConc and ChaDuraLife V1.0 differ in considering "curing age (t ref)" in chloride diffusion theory. • Concrete chloride ion diffusion coefficient relates to "exposure time (t)" via a power function. • In the chloride ion diffusion model, D t 's integral time range is 0∼t, not t ref ∼t ref +t. • ChaDuraLife V1.0 outperforms ClinConc in predicting concrete chloride ion diffusion. [ABSTRACT FROM AUTHOR]

A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic forms and prove the odd-regularities of all but six candidates. [ABSTRACT FROM AUTHOR]

Let r Q (n) be the representation number of a nonnegative integer n by an integral positive definite quadratic form Q in 2 k variables. Let N be the level of Q. For a fixed prime p ∤ N , we assume that r Q (n) satisfies the relation r Q (1) r Q (p 2 n) = r Q (p 2) r Q (n) for all positive integers n with p ∤ n. We actually show that this relation holds for any prime p ∤ N when (− 1) k N is a fundamental discriminant. [ABSTRACT FROM AUTHOR]

In nature and the technology world, acquired signals and time series are usually affected by multiple complicated factors and appear as multi-component non-stationary modes. In many situations it is necessary to separate these signals or time series to a finite number of mono-components to represent the intrinsic modes and underlying dynamics implicated in the source signals. Recently the synchrosqueezed transform (SST) was developed as an empirical mode decomposition (EMD)-like tool to enhance the time-frequency resolution and energy concentration of a multi-component non-stationary signal and provides more accurate component recovery. To recover individual components, the SST method consists of two steps. First the instantaneous frequency (IF) of a component is estimated from the SST plane. Secondly, after IF is recovered, the associated component is computed by a definite integral along the estimated IF curve on the SST plane. The reconstruction accuracy for a component depends heavily on the accuracy of the IFs estimation carried out in the first step. More recently, a direct method of the time-frequency approach, called signal separation operation (SSO), was introduced for multi-component signal separation. While both SST and SSO are mathematically rigorous on IF estimation, SSO avoids the second step of the two-step SST method in component recovery (mode retrieval). The SSO method is based on some variant of the short-time Fourier transform. In the present paper, we propose a direct method of signal separation based on the adaptive continuous wavelet-like transform (CWLT) by introducing two models of the adaptive CWLT-based approach for signal separation: the sinusoidal signal-based model and the linear chirp-based model, which are derived respectively from sinusoidal signal approximation and the linear chirp approximation at any time instant. A more accurate component recovery formula is derived from linear chirp local approximation. We present the theoretical analysis of our approach. For each model, we establish the error bounds for IF estimation and component recovery. [ABSTRACT FROM AUTHOR]