Abstract: Cloud-based enterprise resource planning (ERP) is a combination of standard ERP system and cloud flexibility. Adopting a suitable cloud-based ERP system is an uncertain multi-criteria decision-making problem. Probabilistic linguistic term sets (PLTSs) can be used to address the uncertainty in such problem. On the basis of PLTSs, we introduce an innovative two-step comparative method for the evaluation of cloud-based ERP systems. The extended multiple multi-objective optimisations by ratio analysis method within PLTSs are proposed to obtain the class of cloud-based ERP vendors. Two classes, namely, accepted and rejected, are obtained. Then, we select the suitable cloud-based ERP packages in the accepted class. We put forward the probabilistic linguistic Choquet integral operator to aggregate the ERP package evaluation matrices given the interrelationships between criteria. Subsequently, a new comparison method is introduced to obtain the final results. We conduct an illustrative example to prove the rationality of the presented method. The feasibility of this model is verified by comparative analysis and validity test.

Abstract: This study aims to introduce a series of novel distance measures of an intuitionistic fuzzy set and an extended Multi-attributive Border Approximation Area Comparison method to address multi-criteria group decision-making problems. To aggregate the intuitionistic fuzzy information, we propose two aggregation operators, namely, the intuitionistic fuzzy Dombi generalised $$ \lambda $$-Shapley Choquet arithmetical average operator and intuitionistic fuzzy Dombi generalised $$ \lambda $$-Shapley Choquet geometric average operator. These aggregation operators consider the importance of combinations and correlations among combinations of input arguments. Furthermore, an illustrative example of a human resource management problem is presented to verify the effectiveness of the proposed method, and sensitivity and comparison analyses are conducted to demonstrate the technique’s stability and advancements. Finally, conclusions are drawn.

Abstract: We present a new numerical method for solving fractional delay differential equations (FDDEs) along with its error analysis. We illustrate applicability and utility of the method by solving various examples. Further, we compare the method with other existing methods such as fractional Adams method (FAM) and new predictor–corrector method (NPCM) developed by Daftardar-Gejji et al. (Fract Calc Appl Anal 18(2):400–418, 2015). The order of accuracy is shown to be $$O(h^2).$$ It is noted that the new method is more time efficient and works for very small values of the order of the fractional derivative, where FAM as well as NPCM fail.

Abstract: Hesitant fuzzy set is a powerful and effective tool to express uncertain information in multi-attribute decision-making (MADM) process, as it permits the membership degree of an element to a set represented by several possible values in [0,1]. In this paper, we develop a new decision-making approach based on graph theory to deal with the MADM problems, in which the decision information is expressed by hesitant fuzzy elements. Meanwhile, we generalize this approach to make it suitable for processing interval-valued hesitant fuzzy and hesitant triangular fuzzy information. Moreover, we utilize the numerical examples concerning the energy project selection and software evaluation to show the detailed implementation procedure and reliability of our method in solving MADM problems under hesitant fuzzy, interval-valued hesitant fuzzy and hesitant triangular fuzzy environment.

Abstract: In this paper, our interest is in investigating the monotone inclusion problems in the framework of real Hilbert spaces. To solve this problem, we propose a new modified forward–backward splitting method using the viscosity method (Moudafi in J Math Anal Appl 241(527):46–55, 2000). Under some mild conditions, we establish the strong convergence of the iterative sequence generated by the proposed algorithm. The advantage of our algorithm is that it does not require the co-coercivity of the single-valued operator. Our result improves related results in the literature. Finally, the performances of our proposed method are presented through numerical experiments in signal recovery.

Abstract: This paper proposes a new delayed stochastic epidemic model with double epidemic hypothesis and a general nonlinear response function. The main purpose is to explore the effects of environmental noise and the general nonlinear response function on stochastic dynamics. By constructing appropriate Lyapunov functions and using some novel differential inequality techniques, we first investigate the long-time asymptotic properties of the stochastic delayed system. Moreover, the threshold conditions for the persistence in mean are established. At last, we carry out a series of numerical simulations to illustrate the performance of the theoretical results. The developed theoretical methods and stochastic inequalities techniques can be applied to explore stochastic differential systems with the general response function.

Abstract: The objective of this work is to discuss a novel nonlinear-programming (NLP) models based on the “Technique for Order Preference with respect to the Similarity to the Ideal Solution (TOPSIS)” method to solve the decision-making problems under the cubic intuitionistic fuzzy sets (CIFSs) environment. In the existing studies, the information related to an element is collected either as an interval-valued intuitionistic fuzzy sets (IVIFSs) or intuitionistic fuzzy sets (IFSs) information. An alternative to this, CIFS is one of the extensions of these sets where the information is gathered by considering both the IVIFS and IFS simultaneously. Thus, motivated by this, we modeled the NLP models by considering the interval weights as well as the concept of the relative closeness coefficient and weighted distance measures. Some of the salient features of the models are also examined. Furthermore, we present a novel multicriteria decision-making (MCDM) method and illustrate with a real-life example related to signal processing in sound navigation and ranging. A comparative analysis is also conducted to verify effectiveness and rationality of the method.

Abstract: In this work, the Caputo q-fractional initial value problem of order $$\alpha \in (0,1)$$ in uncertain environment is investigated. The uncertainties are considered as possibility sets using fuzzy sets. The concepts of horizontal membership function and granular difference are used to give a new definition for fuzzy fractional integral and derivative, called granular Riemann–Liouville q-fractional integral and granular Caputo q-fractional derivative. The existence and uniqueness of solution for the Caputo q-fractional initial value problem under granular differentiability concept is established. Finally, some illustrative examples are also presented.

Abstract: We consider the nonlinear Cauchy problem for $$ \varPsi $$ -Hilfer fractional differential equations and investigate the existence, interval of existence and uniqueness of solution in the weighted space of functions. The continuous dependence of solutions on initial conditions is proved via Weissinger fixed point theorem. Picard’s successive approximation method has been developed to solve nonlinear Cauchy problem for differential equations with $$ \varPsi $$ -Hilfer fractional derivative and an estimation has been obtained for the error bound. Further, by Picard’s successive approximation, we derive the representation formulae for the solution of linear Cauchy problem for $$ \varPsi $$ -Hilfer fractional differential equation with constant coefficient and variable coefficient in terms of Mittag-Leffler function and Generalized (Kilbas–Saigo) Mittag-Leffler function respectively.

Abstract: In this paper, by combining the operator splitting technique, a new mass-conserved domain decomposition method for two-dimensional heat equations is proposed. Along the each direction, the interface fluxes are first calculated from the explicit fluxes, then the sub-domain’s interior solutions are paralelly computed by the C–N implicit scheme. The scheme is stable under the condition $$r\le 2(\sqrt{6}-2)$$ and the corresponding convergence order of the scheme are given in $$L^2$$ -norm. Numerical results confirm the theoretical results.

Abstract: This paper proposes a new method for solving distributed order time-fractional reaction–diffusion equations (DO-TFRDEs). Extended versions of the shifted Jacobi–Gauss–Lobatto and shifted fractional order Jacobi–Gauss–Radau collocation methods are developed for reducing the DO-TFRDEs to systems of algebraic equations and computing their approximate solutions. The applicability and accuracy of the method is illustrated through numerical examples.

Abstract: In the current era, most of the researchers addressed an issue while dealing with uncertainty and indeterminacy that exists in fuzzy attributes. It becomes more complex when indeterminacy exists independently when compared to acceptation and rejection part. Due to which, some of the researchers tried to develop a three-way fuzzy concept lattice using neutrosophic set for characterization of uncertainty based on its acceptation, rejection, and uncertain parts, independently. In this process, a problem was addressed while processing the neutrosophic context based on user-required subset of attributes. It takes more time to extract interesting pattern from a given neutrosophic context having a large number of attributes. One of the solutions is to decompose the neutrosophic context via a defined multi-granulation for the truth, falsity and indeterminacy, membership values. To accomplish this task, a method is proposed in this paper using the computing paradigm of granular computing and applied lattice theory with an example.

Abstract: This paper presents a new numerical approximation method to solve a system of nonlinear Fredholm integral equations of second kind. Spectral collocation method and their properties are applied to determine the general solution procedure for nonlinear Fredholm integral equations (FIEs). The convergence and error analysis of spectral collocation method are incorporated for the given nonlinear model. Legendre–Gauss–Lobatto (LGL) points are used as collocation points with various Legendre–Gauss quadrature with weight functions. The use of Legendre polynomials, together with the Gauss quadrature collocation points is well known for the accurate approximations that converge exponentially. Finally, we validate our theoretical results with a number of numerical examples, which further enhance the efficiency of our proposed scheme.

Abstract: In this paper a multistep collocation method for solving Volterra integral equations of the third kind is explained and analyzed. The structure of the method, its solvability and convergence analysis are investigated. Moreover to show the applicability of the presented method and to confirm our theoretical results some numerical examples are given.

Abstract: In this article, a variable-order operational matrix of Gegenbauer wavelet method based on Gegenbauer wavelet is applied to solve a space–time fractional variable-order non-linear reaction–diffusion equation and non-linear Galilei invariant advection diffusion equation for different particular cases. Operational matrices for integer-order differentiation and variable-order differentiation have been derived. Applying collocation method and using the said matrices, fractional-order non-linear partial differential equation is reduced to a system of non-linear algebraic equations, which have been solved using Newton iteration method. The salient feature of the article is the stability analysis of the proposed method. The efficiency, accuracy and reliability of the proposed method have been validated through a comparison between the numerical results of six illustrative examples with their existing analytical results obtained from literature. The beauty of the article is the physical interpretation of the numerical solution of the concerned variable-order reaction–diffusion equation for different particular cases to show the effect of reaction term on the pollution concentration profile.

Abstract: The circulant matrices and skew-circulant matrices are two special classes of Toeplitz matrices and play vital roles in the computation of Toeplitz matrices. In this paper, we focus on real circulant and skew-circulant matrices. We first investigate their real Schur forms, which are closely related to the family of discrete cosine transform (DCT) and discrete sine transform (DST). Using those real Schur forms, we then develop some fast algorithms for computing real circulant, skew-circulant, and Toeplitz matrix-real vector multiplications. Also, we develop a DCT-DST version of circulant and skew-circulant splitting (CSCS) iteration for real positive definite Toeplitz systems. Compared with the fast Fourier transform (FFT) version of CSCS iteration, the DCT-DST version is more efficient and saves a half storage. Numerical experiments are presented to illustrate the effectiveness of our method.

Abstract: In the present research article, we used a new numerical technique called Chebyshev wavelet method for the numerical solutions of fractional delay differential equations. The Caputo operator is used to define fractional derivatives. The numerical results illustrate the accuracy and reliability of the proposed method. Some numerical examples presented which have shown that the computational study completely supports the compatibility of the suggested method. Similarly, a proposed algorithm can also be applied for other physical problems.

Abstract: We present a new approach depending on reproducing kernel method (RKM) for time-fractional Kawahara equation with variable coefficient. This approach consists of obtaining an orthonormal basis function on specific Hilbert spaces. In this regard, some special Hilbert spaces are defined. Kernel functions of these special spaces are given and basis functions are obtained. The approximate solution is attained as serial form. Convergence analysis, error estimation and stability analysis are presented after obtaining the approximate solution. To show the power and effect of the method, two examples are solved and the results are given as table and graphics. The results demonstrate that the presented method is very efficient and convenient for Kawahara equation with fractional order.

Abstract: The generalized inverse has many important applications in the aspects of theoretic research of matrices and statistics. One of the core problems in generalized inverse is to find the necessary and sufficient conditions of the forward order laws for generalized inverse of matrix product. In this paper, by using the expressions for maximal ranks of the generalized Schur complement, we obtain some necessary and sufficient conditions for the forward order laws \(A_1\{1,3\}A_2\{1,3\}A_3\{1,3\}\subseteq (A_1A_2A_3)\{1,3\}\) and \(A_1\{1,4\}A_2\{1,4\}A_3\{1,4\}\subseteq (A_1A_2A_3)\{1,4\}\).

Abstract: In this article, we analyze the stability of a time-delayed susceptible–infected–recovered (S–I–R) epidemic model by introducing two explicit treatment classes (or compartments) along with nonlinear incidence rate. The treatment classes are named as a pre-treated class $$ \left( {T_{1} } \right) $$ and post-treated class $$ \left( {T_{2} } \right) $$. The pre-treatment and post-treatment rates are being considered as Holling type I and Holling type III, respectively. Long-term qualitative analysis has been carried out after incorporating incubation time delay $$ \left( \tau \right) $$ into the incidence rate. The model analysis shows that the model has two equilibrium points, named as disease-free equilibrium (DFE) and endemic equilibrium (EE). The disease-free equilibrium is locally asymptotically stable when the basic reproduction number ($$ R_{0} $$) is less than one and unstable when $$ R_{0} $$ is greater than one for time lag $$ \tau \ge 0 $$, and when $$ R_{0} = 1 $$ by Castillo-Chavez and Song theorem, the disease-free equilibrium changes its stability from stable to unstable and the model exhibits transcritical bifurcation. Furthermore, some conditions for stability of the endemic equilibrium are obtained. Finally, numerical simulations are presented to exemplify the analytical studies.

Abstract: In this article, we develop an invariant subspace method for a system of time-fractional nonlinear partial differential equations in $$(1+2)$$ dimensions. Efficacy of the method is demonstrated by solving coupled system of nonlinear time-fractional diffusion equations and coupled system of time-fractional Burger’s equations in higher dimensions. Furthermore, the algorithmic approach to find more than one invariant subspace is proposed and corresponding exact solutions are constructed.

Abstract: In this paper, we will present a variational PDE-based image inpainting model in which we have used the square of the $$L^2$$ norm of Hessian of the image u as regularization term. The Euler–Lagrange equation will lead us to a fourth-order linear PDE. For time discretization, we have used convexity splitting and the resulting semi-discrete scheme is solved in Fourier domain. Stability analysis for the semi-discrete scheme is carried out. We will demonstrate some numerical results and compare with $$\text {TV}-L^2$$ and $$\text {TV}-H^{-1}$$ model.

Abstract: This paper focuses on the numerical solution for Volterra integral equations of the first kind with highly oscillatory Bessel kernel and highly oscillatory triangle function on the right-hand side. We first establish a new existence theorem of solutions for such equations, and then, the explicit formulas of the solution are derived based on Laplace and inverse Laplace transforms. Furthermore, high-order accurate numerical solutions for approximating the explicit solution are further deduced by applying the Clenshaw–Curtis–Filon method and other effective numerical methods. Preliminary numerical results not only show the exact formulas of the solution, but also present the accuracy of the approximations.

Abstract: In this paper, we present a new conjugate gradient method, in which the search direction is computed by minimizing a selected approximate model in a two-dimensional subspace. That is, if the objective function is not close to a quadratic, the search direction is generated by a conic model. Otherwise, a quadratic model is considered. The direction of the proposed method is proved to possess the sufficient descent property. With the modified nonmonotone line search, we establish a global convergence of the proposed method under appropriate assumptions. R-linear convergence of the proposed method is also analyzed. Numerical results using two different test function collections show that the proposed algorithm is efficient.

Abstract: In this work, the (2+1)-dimensional Korteweg-de Vries equation is investigated, which can be used to represent the amplitude of the shallow–water waves in fluids or electrostatic wave potential in plasmas. By employing the properties of Bell’s polynomial, we obtain bilinear representation of the equation with the aid of an appropriate transformation. Based on the obtained Hirota bilinear form, its lump solutions with localized characteristics are constructed in detail. We then derive the lumpoff solutions of the equation by studying a soliton solution generated by lump solutions. Furthermore, special rogue wave solutions with predictability are well presented, and the time and place of appearance are also derived. Finally, some graphic analysis is represented to better understand the propagation characteristics of the obtained solutions. It is hoped that our results provided in this work can be used to enrich the dynamic behaviors of the equation.

Abstract: This present investigation is contemplated to provide Legendre spectral collocation method for solving multi-Pantograph delay boundary value problems (BVPs). In this regard, an equivalent integral form of such BVPs has been considered. The proposed method is based on Legendre–Gauss collocation nodes and Legendre–Gauss quadrature rule. Convergence analysis associated to the presented scheme has been provided to show its applicability theoretically. Some numerical examples are given to demonstrate the efficiency, accuracy, and versatility of our method. Numerical results confirm the theoretical predictions and are superior with respect to several recent numerical methods including Hermite collocation approach, Laguerre collocation technique and the reproducing kernel method.

Abstract: This paper addresses the approximate solution of the fractional Riccati differential equation (FRDE) in large domains. First, the solution interval is divided into a finite number of subintervals. Then, the Legendre–Gauss–Radau points along with the Lagrange interpolation method are employed to approximate the FRDE solution in each subinterval. The method has the advantage of providing the approximate solutions in large intervals. Additionally, the convergence analysis of the numerical algorithm is also provided. Three illustrative examples are given to illustrate the efficiency and applicability of the proposed method.

Abstract: In this study, first we derive a novel iteration scheme for the sign of a matrix with no pure imaginary eigenvalues. The fourth-order convergence speed of this scheme is given in detail. Secondly, we extend the obtained results so as to calculate the solution of the Yang–Baxter-like equation for the matrix A with no pure imaginary eigenvalues. Some numerical tests are also furnished to manifest the applicability of our method.

Abstract: In this paper, we treat a numerical scheme for the regular fractional Sturm–Liouville problem containing the Prabhakar fractional derivatives with the mixed boundary conditions. We show that the eigenfunctions corresponding to distinct numerical eigenvalues are orthogonal in the Hilbert spaces. The numerical errors and convergence rates are also investigated. Further, we consider a space-fractional diffusion equation and study the associated fractional Sturm–Liouville problem along with the convergence analysis.