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The sun is a fundamental element of the natural environment, and kinetic equations are crucial mathematical models for determining how quickly the chemical composition of a star like the sun is changing. Taking motivation from these facts, we develop and solve a novel fractional kinetic equation containing Fermi–Dirac (FD) and Bose–Einstein (BE) functions. Several distributional properties of these functions and their proposed new generalizations are investigated in this article. In fact, it is proved that these functions belong to distribution space  D ′ while their Fourier transforms belong to  Z ′. Fourier and Laplace transforms of these functions are computed by using their distributional representation. Thanks to them, we can compute various new fractional calculus formulae and a new relation involving the Fox–Wright function. Some fractional kinetic equations containing the FD and BE functions are also formulated and solved. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of many solutions a degenerate weighted elliptic equation which its the number of solutions depend on degeneracy term a i.e., the number of subdomains of Ω\a−1(0) whose boundary is made by submanifolds with 1-codimension. Moreover, by the Ljusternik-Schnirelman principle, we would study the corresponding eigenvalue problem and obtained the nonnegative eigenvalues sequence of main problem. Finally, we will discuss of the regularity of the solutions on C1,α (Ω). [ABSTRACT FROM AUTHOR]

To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method's effectiveness and accuracy. [ABSTRACT FROM AUTHOR]

Mathematics plays an important role in the linguistic structure of artificial intelligence (AI). We describe the linguistic process as a unique structure present both in human cognition and in cognitive computing. The close relationship with both AI and human cognition is due to this unique structure, which paves the way for AI to interfere with the behavior of those who interact with it. We highlight the role of mathematicians in designing algorithms--the core of the AI linguistic process--and in defining steps and instructions for AI. Because algorithms, through AI, interfere with the thought of those who interact with AI by providing anticipated solutions that prevent users from making free choices, we investigate how we can apply ethical principles to guide interactions between users and intelligent systems in order to address this issue. We contend that by integrating ethical principles in the mathematical modeling of algorithms, we can avoid manipulation, inequality, and black boxes in the protection of individual rights. As such, ethical considerations are highly important for those working with algorithms, which highlights the humanistic side of mathematics. [ABSTRACT FROM AUTHOR]

The semantic web invests in systems that work collaboratively. In this article we show that the collaborative way is not enough, because the system must 'understand' the data resources that are provided to it, to organize them in the direction indicated by the system's core, the algorithm. In order for intelligent systems to imitate human cognition, in addition to technical skills to model algorithms, we show that the specialist needs a good knowledge of the principles that explain how human language constructs concepts. The content of this article focuses on the principles of the conceptual formation of language, pointing to aspects related to the environment, to logical reasoning and to the recursive process. We used the strategy of superimposing the dynamics of human cognition and intelligent systems to open new frontiers regarding the formation of concepts by human cognition. The dynamic aspect of the recursion of the human linguistic process integrates visual, auditory, tactile input stimuli, among others, to the central nervous system, where meaning is constructed. We conclude that the human linguistic process involves axiomatic (contextual/biological) and logical principles, and that the dynamics of the relationship between them takes place through recursive structures, which guarantee the construction of meanings through long-range correlation under scale invariance. Recursion and cognition are, therefore, interdependent elements of the linguistic process, making it a set of sui generis structures that evidence that the essence of language, whether natural or artificial, is a form and not a substance. [ABSTRACT FROM AUTHOR]

The fractional calculus of special functions has significant importance and applications in various fields of science and engineering. Here, we aim to find the fractional integral and differential formulas of the extended hypergeometric-type functions by using the Marichev–Saigo–Maeda operators. All the outcomes presented here are of general attractiveness and can yield a number of previous works as special cases due to the high degree of symmetry of the involved functions. [ABSTRACT FROM AUTHOR]

In this paper, the fundamental properties of fractional calculus are discussed with the aim of extending the definition of fractional operators by using wavelets. The Haar wavelet fractional operator is defined, in a more general form, independently on the kernel of the fractional integral. [ABSTRACT FROM AUTHOR]

Fractional calculus (FC) is useful in studying physical phenomena with memory effect. In this paper, a fractional form of Ambartsumian equation is considered utilizing the Caputo fractional derivative. The Heaviside expansion formula in classical calculus (CC) is extended/developed in view of FC. Then, the extended Heaviside expansion formula is applied to obtain the exact solution in a simplest form. Several theorems and lemmas are proved to facilitate the evaluation of the inverse Laplace transform of specific expressions in fractional forms. The exact solution is established in terms of a one-parameter Mittag-Leffler function which is provided for the first time for the Ambartsumian equation in FC. The present solution reduces to the corresponding one in the relevant literature as the fractional order tends to one. Moreover, the convergence of the obtained solution is theoretically proved. Comparisons with another approach in the literature are performed. The advantage of the present analysis over the existing one in the relevant literature is discussed and analyzed. [ABSTRACT FROM AUTHOR]

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique extension principle. We use this system for solving some types of FDEs by solving a series of important examples of FDEs related to many mathematical applications. The quasi-affine bi-framelet-based methods for numerical FDEs show the advantages of using sparse matrices and its accuracy in numerical analysis. [ABSTRACT FROM AUTHOR]

Mathematics, mathematical modeling of real systems, and mathematical and computer methodologies aimed at the qualitative and quantitative study of real physical systems interact in a nontrivial way. This work aims to examine a new class of inverse problems for a fractional partial differential equation with order fractional 0<ρ≤1$$ 0<\rho \le 1 $$, which leads to the spectral problem involving Hermite's differential equation. We introduce proven theorems on the existence and uniqueness of solutions to the current problem. We obtain solutions in the form of series expansion using the Hermite orthogonal basis. Finally, we discuss the convergence analysis of the obtained solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special kind of nonlinear integral equations, Fredholm‐Hammerstein equations. As done before we use an equivalent reformulation, for which collocation methods are more efficient. For the alternative equation, we use a collocation method based on Haar wavelets to approximate the solution. We discuss the convergence and error analysis of the method, showing that, at the collocation nodes, the order of convergence is higher than expected. The paper concludes with numerical examples, which illustrate the good approximations, conclusions, and ideas for future work in this area. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special kind of nonlinear integral equations, Fredholm‐Hammerstein equations. As done before we use an equivalent reformulation, for which collocation methods are more efficient. For the alternative equation, we use a collocation method based on Haar wavelets to approximate the solution. We discuss the convergence and error analysis of the method, showing that, at the collocation nodes, the order of convergence is higher than expected. The paper concludes with numerical examples, which illustrate the good approximations, conclusions, and ideas for future work in this area. [ABSTRACT FROM AUTHOR]

A new fractional-order wavelet basis as generalization of the classical Legendre wavelet is defined. The operational matrices both for derivative and fractional derivative in the sense of Caputo for this fractional-order wavelet are derived. Then, a numerical scheme based on these operational matrices and the typical Tau method is proposed for solving some nonlinear fractional differential equations. Illustrative examples show that the present wavelet Tau method is numerically effective and convenient for solving fractional differential equations. Moreover, the obtained results confirm that, in comparison with the classical Legendre wavelet method, the fractional-order wavelet basis is more efficient and accurate for solving fractional differential equations. Error analysis and convergence of the proposed wavelet method are also provided. [ABSTRACT FROM AUTHOR]

The (3 + 1)‐dimensional Vakhnenko–Parkes mathematical model has a wide range of applications in science and engineering. In this paper, the model studied is investigated and analyzed by using two effective schemes such as sine‐Gordon expansion method and its newly developed version, rational SGEM. Moreover, many novel properties of model studied are extracted in detail. Furthermore, their wave distributions properties and graphs are also plotted under the strain conditions. Interactions of the gravitational potential and high‐frequency wave perturbation properties are also reported in a detailed manner. [ABSTRACT FROM AUTHOR]

A new framework for pricing European currency option is developed in the case where the spot exchange rate follows a subdiffusive fractional Black-Scholes. An analytic formula for pricing European currency call option is proposed by a mean self-financing delta-hedging argument in a discrete time setting. The minimal price of a currency option under transaction costs is obtained as time-step ..., which can be used as the actual price of an option. In addition, we also show that time-step and long-range dependence have a significant impact on option pricing. [ABSTRACT FROM AUTHOR]

In this paper, we will discuss about the invariance of solution set and present the existence and non-existence of the global solutions a class of initial-boundary value problems with dissipative terms is considered for a class of semilinear degenerate hyperbolic equations on the cone Sobolev spaces. First, we will discuss the invariance of some sets corresponding to the problem (1.1) and then, by using a family of potential wells and concavity methods, we obtain existence and non-existence results of global solutions with exponential decay and show the blow-up in finite time of solutions on a manifold with conical singularities. [ABSTRACT FROM AUTHOR]

Magnetic resonance imaging (MRI) is the most sensitive method to detect chronic nervous system diseases such as multiple sclerosis (MS). In this paper, Brownian motion Hölder regularity functions (polynomial, periodic (sine), exponential) for 2D image, such as multifractal methods were applied to MR brain images, aiming to easily identify distressed regions, in MS patients. With these regions, we have proposed an MS classification based on the multifractal method by using the Self-Organizing Map (SOM) algorithm. Thus, we obtained a cluster analysis by identifying pixels from distressed regions in MR images through multifractal methods and by diagnosing subgroups of MS patients through artificial neural networks. [ABSTRACT FROM AUTHOR]

We construct a continuous linear multistep method with trigonometric coefficients from which a symmetric main method as well as additional methods are reproduced. The main and additional methods whose coefficients depend on the frequency and step length are then applied as a trigonometric symmetric boundary value method (SBVM) to solve systems of second-order initial and boundary value problems of the form y" = f (x, y) without first reducing the ordinary differential equation into an equivalent first-order system. Moreover, the method is successfully applied to solve hyperbolic and elliptic partial differential equations, such as the sine- Gordon and the Poisson equations. The stability property of the SBVM is discussed and numerical experiments are performed to show the accuracy of the method. [ABSTRACT FROM AUTHOR]

We discuss a spectral asymptotics theory of an even zonal metric and a Schrödinger operator with zonal potentials on a sphere. We decompose the eigenvalue problem into a series of one-dimensional problems. We consider the individual behavior of this series of one-dimensional problems. We find certain Weyl's type of asymptotics on the eigenvalues. [ABSTRACT FROM AUTHOR]

In this paper, the distribution of primes and prime-indexed primes (PIPs) is studied by mapping primes into a binary image which visualizes the distribution of primes. These images show that the distribution of primes (and PIPs) is similar to a Cantor dust, moreover the self-similarity with respect to the order of PIPs (already proven in Batchko (2014)) can be seen as an invariance of the binary images. The index of primes plays the same role of the scale for fractals, so that with respect to the index the distribution of prime-indexed primes is characterized by the self-similarity alike any other fractal. In particular, in order to single out the scale dependence, the PIPs fractal distribution will be evaluated by limiting to two parameters, fractal dimension ( δ ) and lacunarity ( λ ), that are usually used to measure the fractal nature. Because of the invariance of the corresponding binary plots, the fractal dimension and lacunarity of primes distribution are invariant with respect to the index of PIPs. [ABSTRACT FROM AUTHOR]

In this article, we investigate the local fractional 3-D compressible Navier-Stokes equation via local fractional derivative. We use the Cantor-type cylindrical co-ordinate method to transfer 3-D compressible Navier-Stokes equation from the Cantorian co-ordinate system to the Cantor-type cylindrical co-ordinate system. [ABSTRACT FROM AUTHOR]

The study of the complex model associated with chaotic models is always the most complicated and fundamental in the current scientific environment. The primary goal of the current paper is to provide an illustration of the fundamental theory while analysing dynamical systems and validating the chaotic behaviour of the Lorenz–Haken (LH) equations, a system of fractional differential equations. The LH equations are used to describe the 4D chaotic laser. The Adams–Bashforth numerical method is used to extract the numerical solutions projected model. The classical model introduces the bifurcation of the parameter linked with the system. The system's uniqueness and existence are confirmed using the fixed-point hypothesis with the Caputo–Fabrizio fractional operator, followed by boundedness and dynamical analysis. Furthermore, the chaotic character of the numerical solutions with different orders obtained at different beginning circumstances is presented. [ABSTRACT FROM AUTHOR]

In this article, the local fractional Homotopy perturbation method is utilized to solve the non-homogeneous heat conduction equations. The operator is considered in the sense of the local fractional differential operator. Comparative results between non-homogeneous and homogeneous heat conduction equations are presented. The obtained result shows the non-differentiable behavior of heat conduction of the fractal temperature field in homogeneous media. [ABSTRACT FROM AUTHOR]

In this paper, we derive some classical and fractional properties of the r R s matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials. [ABSTRACT FROM AUTHOR]

The multifractal analysis of binary images of DNA is studied in order to define a methodological approach to the classification of DNA sequences. This method is based on the computation of some multifractality parameters on a suitable binary image of DNA, which takes into account the nucleotide distribution. The binary image of DNA is obtained by a dot-plot (recurrence plot) of the indicator matrix. The fractal geometry of these images is characterized by fractal dimension (FD), lacunarity, and succolarity. These parameters are compared with some other coefficients such as complexity and Shannon information entropy. It will be shown that the complexity parameters are more or less equivalent to FD, while the parameters of multifractality have different values in the sense that sequences with higher FD might have lower lacunarity and/or succolarity. In particular, the genome of Drosophila melanogaster has been considered by focusing on the chromosome 3r, which shows the highest fractality with a corresponding higher level of complexity. We will single out some results on the nucleotide distribution in 3r with respect to complexity and fractality. In particular, we will show that sequences with higher FD also have a higher frequency distribution of guanine, while low FD is characterized by the higher presence of adenine. [ABSTRACT FROM AUTHOR]

In this article, we study the fractional SIR epidemic model with the Atangana–Baleanu–Caputo fractional operator. We explore the properties and applicability of the ZZ transformation on the Atangana–Baleanu–Caputo fractional operator as the ZZ transform of the Atangana–Baleanu–Caputo fractional derivative. This study is an application of two power methods. We obtain a special solution with the homotopy perturbation method (HPM) combined with the ZZ transformation scheme; then we present the problem and study the existence of the solution, and also we apply this new method to solving the fractional SIR epidemic with the ABC operator. The solutions show up as infinite series. The behavior of the numerical solutions of this model, represented by series of the evolution in the time fractional epidemic, is compared with the Adomian decomposition method and the Laplace–Adomian decomposition method. The results showed an increase in the number of immunized persons compared to the results obtained via those two methods. [ABSTRACT FROM AUTHOR]

The one-dimensional heat equations with the heat generation arising in fractal transient conduction associated with local fractional derivative operators are investigated. Analytical solutions are obtained by using the local fractional Adomian decomposition method via local fractional calculus theory. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique. [ABSTRACT FROM AUTHOR]

This paper deals with the complex unit roots representation of archea DNA sequences and the analysis of symmetries in the wavelet coefficients of the digitalized sequence. It is shown that even for extremophile archaea, the distribution of nucleotides has to fulfill some (mathematical) constraints in such a way that the wavelet coefficients are symmetrically distributed, with respect to the nucleotides distribution. [ABSTRACT FROM AUTHOR]

A hybrid model, on the competition tumor cells immune system, is studied under suitable hypotheses. The explicit form for the equations is obtained in the case where the density function of transition is expressed as the product of separable functions. A concrete application is given starting from a modified Lotka-Volterra system of equations. [ABSTRACT FROM AUTHOR]

Abstract: The competitive interaction between the Immune System and tumours is very complex, being non-linear and, to some extent, evolutionary. A fundamental aspect of this evolution is the asynchronous process of mutual learning of the two populations involved — the tumoural and the immune cells. In this work, to describe them, we propose a simple non-linear family of super-macroscopic models with non-monotonically time-varying coefficients. Numerical simulations of transitory phases complement the theoretical analysis. [Copyright &y& Elsevier]

The multiscale (wavelet) decomposition of the solution is proposed for the analysis of the Poisson problem. The approximate solution is computed with respect to a finite dimensional wavelet space [4, 5, 7, 8, 16, 15] by using the Galerkin method. A fundamental role is played by the connection coefficients [2, 7, 11, 9, 14, 17, 18], expressed by some hypergeometric series. The solution of the Poisson problem is compared with the approach based on Daubechies wavelets [18]. [ABSTRACT FROM AUTHOR]

The article is devoted to the description of the results of wavelet analysis of fetal heart rate detecting by cardiotocography method. A number of conclusions are made on the base of such an analysis. It is a part of the research program of creation of a new diagnostic model estimating fetal conditions in antepartum period. [ABSTRACT FROM AUTHOR]

Abstract: Different membrane manufacturer software are available for the designers of RO plants. Software releases are reissued periodically and distributed to the designers to make simulation of membrane behaviour. If you are familiar with these software you know that, a part the different products characteristics, depending on the membrane specification, the acid dosage requests to modify the pH of the feed water are very different for each supplier and often also for each software release. In the paper, using a data input a typical sea water Mediterranean Sea analysis, the membrane software are used to simulate different design conditions and the acid dosage figures are compared with those obtained from an original model developed by the authors, based on the carbonate salts equilibrium. [Copyright &y& Elsevier]

Our object of interest is nonlinear interaction of waves in elastic materials. The new model of a material is proposed that takes into account the mechanism of simultaneous quadratic and cubic nonlinear deformations. Introduction of cubic nonlinearity into the model makes the general wave picture more complicated and creates new possibilities for the wave analysis. We present four possibilities for the evolution of profiles of plane harmonic waves. It is noted that quadratic and cubic nonlinearities emerge first of all in the second and third harmonics generation, respectively. Further, we discuss the results of computer modelling of the wave profile evolution. The influence of the progress of second and third harmonics on the wave profile evolution is studied separately. We study separately how second and third harmonics influence the evolution of the wave profile. We also investigate how the progress of harmonics depends on the initial frequency and amplitude. We find two distinct schemes of the evolution progress: the scheme (in) with four stages for the second harmonics and the scheme with three stages for the third harmonics. As a result the influence of both harmonics could be observed simultaneously, and such a case is demonstrated in the paper. Nevertheless this phenomenon is not necessarily present in every material which explains the absence of experimental observations of the third harmonics by this time. [ABSTRACT FROM AUTHOR]

The compression property of wavelets in the analysis of an evolution problem (with unsmooth initial conditions) is investigated. The effectiveness of wavelets both in the reduction of complexity (number of coefficients) and in better approximation is shown. Haar wavelets, having the simplest interpretation of the wavelet coefficients, are used for defining the wavelet solution of an evolution (parabolic-hyperbolic) problem. The approximate solution, at a given fixed scale (resolution), results from the superimposition of (a small set of) fundamental wavelets, thus giving (also) a physical interpretation to wavelets. Since Haar wavelets are not smooth enough, a numerical derivative algorithm, which allows the scale approximation of partial differential evolution operators, is also defined. As application, the heat propagation (of an initial square wave) is explicitly given in terms of wavelets. [ABSTRACT FROM AUTHOR]

We propose two efficient numerical algorithms, Bernoulli uniform collocation method and Bernoulli Chebyshev collocation method for solving 3 r d -order Lane-Emden-Fowler boundary value problems. To avoid singularity at x = 0 , we transform the concerned problem into its integral form. By using the Bernoulli collocation method, the resulting integral equation is converted into a system of nonlinear equations to be solved numerically by any suitable iteration method. The error bound of the algorithm and the existence of a unique solution of the problem are also discussed. The high accuracy and efficiency of the proposed method are shown by comparing the results of L ∞ and L 2 errors of several examples. Moreover, the numerical results of our method are compared with the results obtained by the other known techniques. [ABSTRACT FROM AUTHOR]

The aim of this work is to consider bicomplex Bernstein polynomials attached to analytic functions on a compact C 2 -disk and to present some approximation properties extending known approximation results for the complex Bernstein polynomials. Furthermore, we obtain and present quantitative estimate inequalities and the Voronovskaja-type result for analytic functions by bicomplex Bernstein polynomials. [ABSTRACT FROM AUTHOR]

Capturing solution near the singular point of any nonlinear SBVPs is challenging because coefficients involved in the differential equation blow up near singularities. In this article, we aim to construct a general method based on orthogonal polynomials as wavelets, i.e., orthogonal polynomial wavelet method (OPWM). We also discuss the convergence of OPWM as a particular case. We discuss multiresolution analysis for wavelets generated by orthogonal polynomials, e.g., Hermite, Legendre, Chebyshev, Laguerre, and Gegenbauer. Then we use these orthogonal polynomial wavelets for solving nonlinear SBVPs. These wavelets can deal with singularities easily and efficiently. To deal with the nonlinearity, we use both Newton's quasilinearization and the Newton–Raphson method. To show the importance and accuracy of the proposed methods, we solve the Lane–Emden type of problems and compare the computed solutions with the known solutions. As the resolution is increased the computed solutions converge to exact solutions or known solutions. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to define a simple but effective method for approximating solutions of multi-order fractional differential equations relying on Caputo fractional derivative and under supplementary conditions. Our basis functions are based on some original generalization of the Bessel polynomials, which satisfy many properties shared by the classical orthogonal polynomials as given by Hermit, Laguerre, and Jacobi. The main advantages of our polynomials are two-fold: All the coefficients are positive and any collocation matrix of Bessel polynomials at positive points is strictly totally positive. By expanding the unknowns in a (truncated) series of basis functions at the collocation points, the solution of governing differential equation can be easily converted into the solution of a system of algebraic equations, thus reducing the computational complexities considerably. Several practical test problems also with some symmetries are given to show the validity and utility of the proposed technique. Comparisons with available exact solutions as well as with several alternative algorithms are also carried out. The main feature of our approach is the good performance both in terms of accuracy and simplicity for obtaining an approximation to the solution of differential equations of fractional order. [ABSTRACT FROM AUTHOR]

Power load fluctuation is generally agreed to be a non-stationary stochastic process. The Fractional Brownian Motion (FBM) model is proposed to forecast a non-stationary time series with high accuracy. Computation of the Hurst exponent (H) for the power load data series using the Rescaled Range Analysis (R/S) in this study. This method is used to verify the Long-Range Dependent (LRD) characteristics of non-stationary power load data. For the real power load, however, H exponent takes on the self-similarity characteristics in a certain finite range of intervals, the global self-similarity is very rare to exist. The H exponent of the self-similarity usually has more than one value. We generalize multifractional H (t) to replace constant H. To improve the forecasting accuracy, the H (t) is optimized by the Quantum-Behaved Particle Swarm Optimization (QPSO). Once the optimal H (t) is obtained, then the optimal and parameters in the multi-Fractional Brownian Motion (mFBM) model can be deduced to forecast next power load data series with a higher accuracy. • Fractional Brownian Motion (FBM)as Stochastic model forecasts a power load series. • The parameters of the FBM are estimated by R/S and MSE methods. • The parameters of the FBM are optimized by PSO and QPSO. • The prediction errors of RNN,FBM,PSO + FBM,QPSO + FBM were analyzed by using box-plot and norm-plot. [ABSTRACT FROM AUTHOR]

In this paper, local fractional cylindrical wave solutions on Signorini hyperelastic materials are studied. In particular, we focus on the so-called Signorini potential. Cantor-type cylindrical coordinates are used to analyze, both from dynamical and geometrical point of view, wave solutions, so that the nonlinear fundamental equations of the fractional model are explicitly given. In the special case of linear approximation we explicitly compute the fractional wave profile. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the influence of the choice of continuous linear operator for obtaining the approximate periodic solutions of ordinary second-order differential equations. In most of these problems, the periods are unknown, and the determination of these periods and periodic solutions is a difficult issue. So, a new computational method is proposed based on the symmetric operator, namely the reproducing kernel Hilbert space (RKHS) method to obtain the interval of these solutions. This operator, as a consequence of the symmetric inner product, is a symmetric operator and it will be used to show the influence on periodic solutions. The high efficiency of the proposed strategy is presented along with some illustrative examples which demonstrate their periodic interval dealing with the choice of an appropriate continuous linear operator. [ABSTRACT FROM AUTHOR]

Solving Burgers' equation always posses challenges for a small value of viscosity. Here we present a numerical method based on the Haar wavelet collocation method coupled with a nonstandard finite difference (NSFD) scheme for a class of generalized Burgers' equation. In the solution process, the time derivative is discretized by the NSFD scheme and the spatial derivatives are approximated by the Haar wavelets series. The nonlinear terms are linearized with the help of the quasilinearisation process. We illustrate the efficiency of the proposed method by solving several test problems and report their L 2 -error and L ∞ -error norms. The derived method is quite easy to implement compared to the other methods. Also, the error analysis of the current method is discussed. It is also observed that for the small number of grid points, the current method produces results that are in great agreement with the analytical solutions. [ABSTRACT FROM AUTHOR]

This paper introduces an efficient numerical scheme for solving a significant class of fractional differential equations. The major contributions made in this paper apply a direct approach based on a combination of time discretization and the Laplace transform method to transcribe the fractional differential problem under study into a dynamic linear equations system. The resulting problem is then solved by employing the numerical method of the quadrature rule, which is also a well-developed numerical method. The present numerical scheme, which is based on the numerical inversion of Laplace transform and equal-width quadrature rule is robust and efficient. Some numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework. [ABSTRACT FROM AUTHOR]