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Partial combinatory algebras are algebraic structures that serve as generalized models of computation. In this paper, we study embeddings of pcas. In particular, we systematize the embeddings between relativizations of Kleene's models, of van Oosten's sequential computation model, and of Scott's graph model, showing that an embedding between two relativized models exists if and only if there exists a particular reduction between the oracles. We obtain a similar result for the lambda calculus, showing in particular that it cannot be embedded in Kleene's first model., Comment: 31 pages

One of the main open problems regarding decidability of the existential theory of rings is the analogue of Hilbert's Tenth Problem (HTP) for the ring of entire holomorphic functions in one variable. In the direction of a negative solution, we prove unsolvability of HTP for rings of exponential polynomials. This provides the first known case of HTP for a ring of entire holomorphic functions in one variable strictly containing the polynomials. The technique of proof consists of an interaction between Arithmetic, Analysis, Logic, and Functional Transcendence.

The goal of this paper is to discover modern soliton solutions to long and short-wave interaction system by procedures called extended rational sine-cosine and rational sinh-cosh methods. We assume that the equation has a hypothetical soliton solutions. By reorganizing the resulting equations, we obtain a system of equations. Using Maple software, we get unknown coefficients in the system and writing them in the original equation, we obtain new solition solutions of the equation. The results show that the soliton solutions generated by the method for the long and short-wave interaction system are bright, kink type, bright periodic and dark solutions. We provided 3-D figures to illustrate the solutions. Computational results indicate that the method employed in this paper is superior than some other methods used in the literature to solve the same system equations.

The main objective of this article is to study the Wiener index of zero-divisor graph of the ring of integer modulo n . We present a constructed method to calculate the Wiener index of zero-divisor graph of Z n for any positive integer n .

In this paper we introduce and study the algebra of almost strongly fuzzy bounded operators on fuzzy normed linear spaces. Some connections with the algebra of strongly fuzzy bounded operators are presented. The completeness of these algebras is also established. A characterization of bounded elements of the algebra of almost strongly fuzzy bounded operators is given. We also introduce two fuzzy spectral radii for almost strongly fuzzy bounded operators. For bounded elements of the considered algebra the equality between these fuzzy spectral radii is proved.

In this article, a numerical technique based on polynomials is proposed for the solution of one and two-dimensional time-fractional Burgers equation. First, the given problem is reduced to time discrete form using θ -weighted scheme. Then, with the help of Lucas and Fibonacci polynomials the given PDEs transformed to system of algebraic equations which is easy to solve. The proposed algorithm is validated by solving some numerical examples. Despite this, convergence analysis of the scheme is briefly discussed and verified numerically. The main objective of this paper is to show that polynomial based method is convenient for 1D and 2D nonlinear time-fractional partial differential equations (TFPDEs). Efficiency and performance of the proposed technique are examined by calculating L 2 and L ∞ error norms. Obtained accurate results confirm applicability and efficiency of the method.

In this paper we obtain explicit estimates and existence results on the number of F q -rational solutions of certain systems defined by families of diagonal equations over finite fields. Our approach relies on the study of the geometric properties of the varieties defined by the systems involved. We apply these results to a generalization of Waring's problem and the distribution of solutions of congruences modulo a prime number.

Singularly perturbed systems have found widespread applications in practice. The existing results on singularly perturbed systems mainly focused on the Lyapunov asymptotic stability, which are unable to deal with the cases that the system states cannot exceed a given threshold during a fixed time interval. This paper addresses the finite-time fuzzy control issue for discrete-time nonlinear singularly perturbed systems with input constraints. The aim is to guarantee the boundedness of the states of singularly perturbed systems during a finite-time interval. Based on the matrix inequality technique, some conditions are established to guarantee the finite-time boundedness of the fuzzy singularly perturbed systems, where the singularly perturbed parameter is independent so as to avoid the ill-conditioned problem caused by the small singularly perturbed parameter. The gains of the finite-time fuzzy controller can be obtained by solving some singularly perturbed parameter independent linear matrix inequalities. Finally, the proposed finite-time fuzzy controller design approach for nonlinear singularly perturbed systems is illustrated via the Van der Pol circuit.

This article presents an efficient method to solve of Eikonal equation by the new differential quadrature method. The purpose of this article is to obtain the accuracy and performance method for the Eikonal equation. We introduce a novel grid points to reduce errors and compared them on different issues to accelerate convergence. The accuracy and stability of numerical results for two and three-dimensional of the Eikonal equation are presented. In these numerical examples, we investigate the stability of the proposed method by adding random noises. The new differential quadrature method is a capable method for the some numerical examples and we evaluate the accuracy and performance of our proposed method.

In this research, the finite difference method is used to solve the initial value problem of linear first order Volterra-Fredholm integro-differential equations with singularity. By using implicit difference rules and composite numerical quadrature rules, the difference scheme is established on a Shishkin mesh. The stability and convergence of the proposed scheme are analyzed and two examples are solved to display the advantages of the presented technique.

In this paper we develop a general computer-assisted proof method for periodic solutions to delay differential equations. The class of problems considered includes systems of delay differential equations with an arbitrary number of (forward and backward) delays. When the nonlinearities include nonpolynomial terms we introduce auxiliary variables to first rewrite the problem into an equivalent polynomial one. We then apply a flexible fixed point technique in a space of geometrically decaying Fourier coefficients. We showcase the efficacy of this method by proving periodic solutions in the well-known Mackey–Glass delay differential equation for the classical parameter values.

Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates. We go outside Boolean realm and consider circuits built of any fixed set of gates on an arbitrary large finite domain. From the complexity point of view this is strictly connected with the problems of solving equations (or systems of equations) over finite algebras. The research reported in this work was motivated by a desire to know for which finite algebras $\mathbf A$ there is a polynomial time algorithm that decides if an equation over $\mathbf A$ has a solution. We are also looking for polynomial time algorithms that decide if two circuits over a finite algebra compute the same function. Although we have not managed to solve these problems in the most general setting we have obtained such a characterization for a very broad class of algebras from congruence modular varieties. This class includes most known and well-studied algebras such as groups, rings, modules (and their generalizations like quasigroups, loops, near-rings, nonassociative rings, Lie algebras), lattices (and their extensions like Boolean algebras, Heyting algebras or other algebras connected with multi-valued logics including MV-algebras). This paper seems to be the first systematic study of the computational complexity of satisfiability of non-Boolean circuits and solving equations over finite algebras. The characterization results provided by the paper is given in terms of nice structural properties of algebras for which the problems are solvable in polynomial time., 50 pages

In this paper, a new computationally efficient approach to solve fractional differential equations with Atangana–Baleanu operator is introduced. Controlled Picard’s method is employed for solving a class of fractional differential equations with order 0 α 1 . The proposed approach can cover wide range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for nonlinear differential equations. Two models of fractional Riccati equation are solved to validate and illustrate the accuracy of the new approach. Figures has been used to construct the results obtained from the presented approach. It is shown that the proposed method is efficient, credible, and easy to implement for various related problems in science and engineering.

The main target of our research is to achieve the solution to a variable-order fractional mobile-immobile advection-dispersion equation within the Atangana-Baleanu (AB) fractional operator. This equation is the best tool for modeling dynamical systems. The presented scheme is based on the collocation method and Lagrange polynomials. The given approach uses derivative operational matrices of integer and variable orders. The operational matrix of variable-order derivative in the AB sense is computed based on Lagrange polynomials in the presented study. The key theory of this approach is to convert the corresponding problem to a system of algebraic equations. The solution to the problem under study can be computed by utilizing the mentioned system and the collocation points. The aim of this paper is to show that this technique is easily used to solve fractional partial differential equations with excellent accuracy in results. To demonstrate the accuracy and efficiency of the suggested method, some numerical examples are provided.

In this article, based on polynomial differential inclusions, we propose a heuristic iterative approach for estimating the domains of attraction for nonpolynomial systems. First, we use the fuzzy model to construct a polynomial differential inclusion for the nonpolynomial system, which can be equivalently written as a time-invariant uncertain polynomial system. Then, beginning with an initial inner estimation, we present an iterative approach to enlarge this initial inner estimation by calculating common Lyapunov-like functions. Furthermore, the domains of attraction are estimated by combining this iterative approach with heuristic construction of differential inclusions. In the end, our heuristic iterative approach is implemented with linear semidefinite programming and then tested on some nonpolynomial examples with comparisons to the existing methods in the literature.

Let F be a holomorphic map whose components satisfy some polynomial relations. We present an algorithm for constructing Nash maps locally approximating F, whose components satisfy the same relations., Comment: 44 pages; presentation of the results improved

In the area of symbolic-numerical computation within computer algebra, an interesting question is how “close” a random input is to the “critical” ones. Examples are the singular matrices in linear algebra or the polynomials with multiple roots for Newton's root-finding method. Bounds, sometimes very precise, are known for the volumes over R or C of such neighborhoods of the varieties of “critical” inputs; see the references below. This paper deals with the discrete version of this question: over a finite field, how many points lie in a certain type of neighborhood around a given variety? A trivial upper bound on this number is given by the product (size of the variety) ⋅ (size of a neighborhood of a point). It turns out that this bound is usually asymptotically tight, in particular for the singular matrices, polynomials with multiple roots, and pairs of non-coprime polynomials. The interesting question then is: for which varieties is this bound not asymptotically tight? We show that these are precisely those that admit a shift, that is, where one absolutely irreducible component of maximal dimension is a shift (translation by a fixed nonzero point) of another such component. Furthermore, the shift-invariant absolutely irreducible varieties are characterized as being cylinders over some base variety. Computationally, determining whether a given variety is shift-invariant turns out to be intractable, namely NP-hard even in simple cases.

Acknowledgements: We are grateful to the anonymous reviewer for their helpful corrections that have improved the clarity of our exposition. The second author was supported by a London Mathematical Society Early Career Fellowship at the University of Oxford., Let $A$ be a finite-dimensional algebra over a field of characteristic $p>0$. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective $A$--modules $P$ into those of the torsion submodules of $P$. As an application, we show that blocks of both the classical and quantum Schur algebras $S(2,r)$ and $S_q(2,r)$ are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain $2p^k$ simple modules for some $k$.

Autonomous differential equations of fractional order and non-singular kernel are solved. While solutions can be obtained through numerical, graphical, or analytical solutions, we seek an implicit analytical solution.

Model order reduction is a technique that is used to construct low-order approximations of large-scale dynamical systems. In this paper, we investigate a balancing based model order reduction method for dynamical systems with a linear dynamical equation and a quadratic output function. To this aim, we propose a new algebraic observability Gramian for the system based on Hilbert space adjoint theory. We then show the proposed Gramians satisfy a particular type of generalized Lyapunov equations and we investigate their connections to energy functionals, namely, the controllability and observability. This allows us to find the states that are hard to control and hard to observe via an appropriate balancing transformation. Truncation of such states yields reduced-order systems. Finally, based on $\mathcal H_2$ energy considerations, we, furthermore, derive error bounds, depending on the neglected singular values. The efficiency of the proposed method is demonstrated by means of two semi-discretized partial differential equations and is compared with the existing model reduction techniques in the literature., 26 pages

In this paper we investigate non-linear matroid optimization problems with polynomial objective functions where the monomials satisfy certain monotonicity properties. Indeed, we study problems where the set of non-linear monomials consists of all non-linear monomials that can be built from a given subset of the variables. Linearizing all non-linear monomials we study the respective polytope. We present a complete description of this polytope. Apart from linearization constraints one needs appropriately strengthened rank inequalities. The separation problem for these inequalities reduces to a submodular function minimization problem. These polyhedral results give rise to a new hierarchy for the solution of matroid optimization problems with polynomial objectives. This hierarchy allows to strengthen the relaxations of arbitrary linearized combinatorial optimization problems with polynomial objective functions and matroidal substructures. Finally, we give suggestions for future work.

In this paper, fuzzy linear singular differential equations under so-called granular differentiability are investigated in which the coefficients and initial conditions are as fuzzy numbers. Some new notions such as fuzzy nilpotent matrix, fuzzy linearly independent vectors, fuzzy eigenvectors, rank, index and fuzzy Jordan canonical form of a fuzzy matrix are introduced. Moreover, we compare the proposed method with other ones based on other derivatives, using some examples.

Weighted finite automata over the field of rational numbers and unary alphabets are considered. The notion of a characteristic polynomial is introduced for such automata as a means to provide a decidable necessary and sufficient condition, under which a unary weighted automaton admits a deterministic, i.e., sequential equivalent. The sequentiality problem for univariate rational series is thus proved to be decidable both over the rational numbers and over the integers, confirming a conjecture of S. Lombardy and J. Sakarovitch; its decidability over the nonnegative integers is observed as well. The decision algorithm proposed for these tasks is shown to run in polynomial time. A determinisation algorithm for determinisable unary weighted automata over the rational numbers is also described.

In this paper, we propose a real vector representation of reduced quaternion matrix and study its properties. By using this real vector representation, Moore-Penrose inverse, and semi-tensor product of matrices, we study some kinds of solutions of reduced biquaternion matrix equation (1.1). Several numerical examples show that the proposed algorithm is feasible at last.

This paper deals with estimation of interval controls for interval Linear Time Invariant (LTI) plants using pole placement technique. Generally, the parameters associated with LTI plant system are assumed to be crisp values, but due to the occurrence of error in experimentation, measurement or due to maintenance induced errors, the parameters are imprecise. As such, the LTI problem reduces to interval LTI. The pole placement problem is then reduced to interval algebraic equation viz. interval Diophantine equation using proper transfer function. Further, the interval Diophantine equation is transformed to Interval System of Linear Equations (ISLE) viz. interval Sylvester matrix equation. Initially, using interval arithmetic an interval algebraic approach has been applied to solve the ISLE for computing non-negative or non-positive controls. Then, another approach using Artificial Neural Network (ANN) is proposed for computing the interval controls. Further, algorithms based on sign function and ANN procedures have been discussed. Finally, the efficiency of the proposed algorithms has been verified using different order interval plants.

Some new characterizations of the generalized Moore-Penrose inverse are proposed using range, null space, several matrix equations and projectors. Several representations of the generalized Moore-Penrose inverse are given. The relationships between the generalized Moore-Penrose inverse and other generalized inverses are discussed using core-EP decomposition. The generalized Moore-Penrose matrices are introduced and characterized. One relation between the generalized Moore-Penrose inverse and corresponding nonsingular border matrix is presented. In addition, applications of the generalized Moore-Penrose inverse in solving restricted matrix equations are studied.

We develop an effective algorithm to find a well-approximate solution of a generalized Sylvester-transpose matrix equation where all coefficient matrices and an unknown matrix are rectangular. The algorithm aims to construct a finite sequence of approximated solutions from any given initial matrix. It turns out that the associated residual matrices are orthogonal, and thus, the desire solution comes out in the final step with a satisfactory error. We provide numerical experiments to show the capability and performance of the algorithm.

By employing the properties of centered realizations, we devise a modified version of the dual Riccati equation approach to optimal $\mathcal {H}_{2}$ control for differential-algebraic systems that is guaranteed to be more computationally efficient and numerically accurate than other methods from literature. Moreover, we show that if the optimal controller has an improper transfer function matrix, then all controllers which ensure a finite $\mathcal {H}_{2}$ norm will share this property.

The main object of this paper is to investigate the spectral collocation method for three new models of space fractional Fisher equations based on the exponential decay kernel, for which properties of Chebyshev polynomials are utilized to reduce these models to a set of differential equations. We then numerically solve these differential equations using finite differences, with the resulting algebraic equations solved using Newton 's method. The accuracy of the numerical solution is verified by computing the residual error function. Additionally, the numerical results are compared with other results obtained using the power law kernel and the Mittag-Leffler kernel. The advantage of the present work stems from the use of spectral methods, which have high accuracy and exponential convergence for problems with smooth solutions. The numerical solutions based on Chebyshev polynomials are in remarkably good agreement with numerical solutions obtained using the power law and the Mittag-Leffler kernels. Mathematica was used to obtain the numerical solutions.

In this paper, a class of Clifford-valued higher-order Hopfield neural networks with D operator is studied by non-decomposition method. Except for time delays, all parameters, activation functions and external inputs of this class of neural networks are Clifford-valued functions. Based on Banach fixed point theorem and differential inequality technique, we obtain the existence, uniqueness and global exponential stability of compact almost automorphic solutions for this class of neural networks. Our results of this paper are new. In addition, two examples and their numerical simulations are given to illustrate our results.

In this paper, we developed a computational Haar collocation scheme for the solution of fractional linear integro-differential equations of variable order. Fractional derivatives of variable order is described in the Caputo sense. The given problem is transformed into a system of algebraic equations using the proposed Haar technique. The results are obtained by solving this system with the Gauss elimination algorithm. Some examples are given to demonstrate the convergence of Haar collocation technique. For different collocation points, maximum absolute and mean square root errors are computed. The results demonstrate that the Haar approach is efficient for solving these equations.

In this paper, a numerical method is presented to solve fractional boundary value problems. In fractional calculus, the modelling of natural phenomenons is best described by fractional differential equations. So, it is important to formulate efficient and accurate numerical techniques to solve fractional differential equations. In this article, first, we introduce ψ-shifted Chebyshev polynomials then project these polynomials to formulate ψ-shifted Chebyshev operational matrices. Finally, these operational matrices are used for the solution of fractional boundary value problems. The convergence is analysed. It is observed that solution of non-integer order differential equation converges to corresponding solution of integer order differential equation. Finally, the efficiency and applicability of method is tested by comparison of the method with some other existing methods.

We present a method to compute polynomial conservation laws for systems of partial differential equations ( PDE s). The method only relies on linear algebraic computations and is complete, in the sense it can find a basis for all polynomial fluxes that yield conservation laws, up to a specified order of derivatives and degree. We compare our method to state-of-the-art algorithms based on the direct approach on a few PDE systems drawn from mathematical physics.

By using the operator theory, we establish the Green's function for Caputo fractional differential equation under Sturm-Liouville boundary conditions. The results are new, the method used in this paper will provide some new ideas for the study of this kind of problems and easy to be generalized to solving other problems.

The present work is to solve the nonlinear singular models using the framework of the stochastic computing approaches. The purpose of these investigations is not only focused to solve the singular models, but the solution of these models will be presented to the extended form of the delayed, prediction and pantograph differential models. The Gudermannian function is designed using the neural networks optimized through the global scheme "genetic algorithms (GA)", local method "sequential quadratic programming (SQP)" and the hybridization of GA-SQP. The comparison of the singular equations will be presented with the exact solutions along with the extended form of delayed, prediction and pantograph based on these singular models. Moreover, the neuron analysis will be provided to authenticate the efficiency and complexity of the designed approach. For the correctness and effectiveness of the proposed approach, the plots of absolute error will be drawn for the singular delayed, prediction and pantograph differential models. For the reliability and stability of the proposed method, the statistical performances "Theil inequality coefficient", "variance account for" and "mean absolute deviation'' are observed for multiple executions to solve singular delayed, prediction and pantograph differential models.

In this paper, a computational approach based on the operational matrices in conjunction with orthogonal shifted Legendre polynomials (OSLPs) is designed to solve numerically the multi-order partial differential equations of fractional order consisting of mixed partial derivative terms. Our computational approach has ability to reduce the fractional problems into a system of Sylvester types matrix equations which can be solved by using MATLAB builtin function lyap(.). The solution is approximated as a basis vectors of OSLPs. The efficiency and the numerical stability is examined by taking various test examples.

Constrained solutions are required in some practical problems. This paper studies the problem of different constrained solutions for a class of multivariate quadratic matrix equations, which has rarely been studied before. First, the quadratic matrix equations are transformed into linear matrix equations by Newton's method. Second, the different constrained solutions of the linear matrix equations are solved by the modified conjugate gradient method, and then the different constrained solutions of the multivariate quadratic matrix equations are obtained. Finally, the convergence of the algorithm is proved. The algorithm can be well performed on the computers, and the effectiveness of the method is verified by numerical examples.

Let F q denote the finite field with q elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

Differential-algebraic equations (DAEs) are widely used for modeling of dynamical systems. In numerical analysis of DAEs, consistent initialization and index reduction are important preprocessing prior to numerical integration. Existing DAE solvers commonly adopt structural preprocessing methods based on combinatorial optimization. Unfortunately, the structural methods fail if the DAE has numerical or symbolic cancellations. For such DAEs, methods have been proposed to modify them to other DAEs to which the structural methods are applicable, based on the combinatorial relaxation technique. Existing modification methods, however, work only for a class of DAEs that are linear or close to linear. This paper presents two new modification methods for nonlinear DAEs: the substitution method and the augmentation method. Both methods are based on the combinatorial relaxation approach and are applicable to a large class of nonlinear DAEs. The substitution method symbolically solves equations for some derivatives based on the implicit function theorem and substitutes the solution back into the system. Instead of solving equations, the augmentation method modifies DAEs by appending new variables and equations. The augmentation method has advantages that the equation solving is not needed and the sparsity of DAEs is retained. It is shown in numerical experiments that both methods, especially the augmentation method, successfully modify high-index DAEs that the DAE solver in MATLAB cannot handle., Comment: A preliminary version of this paper is to appear in Proceedings of the 44th International Symposium on Symbolic and Algebraic Computation (ISSAC 2019), Beijing, China, July 2019

This research is a development of multiplicative calculus. This study is about the Fourier multiplicative transformation and its application to the multiplicative differential equation. This study aims to determine the Fourier multiplicative transformation as well as the multiplicative differential equation. This study contains numerical simulations to solve the problem of ordinary multiplicative differential equations of the first order. The methods used in this research are descriptive research methods through the study of literature. The results of this study are the application of multiplicative Fourier transformations to multiplicative differential equations and numerical solutions of ordinary multiplicative differential equations with the Adam Bashforth-Moulton multiplicative method. Â Keywords: Multiplicative Calculus, Fourier Multiplicative Transformation, Multiplicative Differential Equations, Adams Bashforth Moulton Multiplicative Method

This paper investigates minimal solutions of fuzzy relation inequalities with addition-min composition. It first shows the conditions that an element is a minimal solution of the inequalities, and presents the conditions that the inequalities have a unique minimal solution. It then proves that every solution of the inequalities has a minimal one and proposes an algorithm to searching for a minimal solution with computational complexity O (n2) where n is the number of unknown variables of the inequalities. This paper finally describes all minimal solutions of the inequalities.

The Jacobi–Davidson iteration method is very efficient in solving Hermitian eigenvalue problems. If the correction equation involved in the Jacobi–Davidson iteration is solved accurately, the simplified Jacobi–Davidson iteration is equivalent to the Rayleigh quotient iteration which achieves cubic convergence rate locally. When the involved linear system is solved by an iteration method, these two methods are also equivalent. In this paper, we present the convergence analysis of the simplified Jacobi–Davidson method and present the estimate of iteration numbers of the inner correction equation. Furthermore, based on the convergence factor, we can see how the accuracy of the inner iteration controls the outer iteration.

We extend the construction of upsilon-type invariants to null-homologous knots in rational homology three-spheres. By considering $m$-fold cyclic branched covers with $m$ a prime power, this extension provides new knot concordance invariants $\Upsilon_m^C (K)$ of knots in $S^3$. We give computations of these invariants for some families of alternating knots and reprove some independence results in the smooth concordance group., Comment: 26 pages, 13 figures. New version fixes typos, minor mistakes and improves readability. Comments are welcome!

The goal of this paper is to generalize and refine some previous inequalities between the LP- norms of the sth derivative and of the polynomial itself, in the case when the zeros are outside of the open unit disk.

Skewes’ number was discovered in 1933 by South African mathematician Stanley Skewes as upper bound for the first sign change of the difference π (x) − li(x). Whether a Skewes’ number is an integer is an open problem of Number Theory. Assuming Schanuel’s conjecture, it can be shown that Skewes’ number is transcendental. In our paper we have chosen a different approach to prove Skewes’ number is an integer, using lattice points and tangent line. In the paper we acquaint the reader also with prime numbers and their use in RSA coding, we present the primary algorithms Lehmann test and Rabin-Miller test for determining the prime numbers, we introduce the Prime Number Theorem and define the prime-counting function and logarithmic integral function and show their relation.

In this paper, we present an indirect computational procedure based on the truncated second kind Chebyshev wavelets for finding the solutions of Caputo fractional time-invariant linear optimal control systems which the functional cost consists of the finite-time quadratic cost function. Utilizing the operational matrices of second Chebyshev wavelets (SCWs) of the Riemann–Liouville fractional integral, the corresponding linear two-point boundary value problem (TPBVP), obtained from the fractional Euler–Lagrange equations, is reduced to a coupled Sylvester-type matrix equation. An equivalent linear matrix form using the Kronecker product is constructed. The upper bound of the error of the SCWs approximation and the convergence of the proposed method are investigated. Low computational complexity and flexible accuracy are two important superiorities of this approach. Numerical experiments provide satisfactory results compared to the exiting techniques.

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular ‎Fredholm integral equations of the ‎second kind. The kernel is ‎interpolated twice concerning ‎both variables and then is transformed into the product of five ‎matrices; two of them are monomial basis ‎matrices. To isolate the singularity of the kernel, we ‎developed two techniques based on a good choice of ‎different two sets of nodes to be distributed ‎over the integration domain. Each set is specific to one of the ‎kernel arguments so that the kernel ‎values never become zero or imaginary. The significant advantage of the ‎two presented ‎techniques is the ability to gain ‎‎access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the ‎mean of the interpolant solution ‎and the maximum error norm estimation are studied. The ‎interpolate solutions of the illustrated four ‎examples are found strongly converging uniformly to the ‎exact solutions.

We propose numerical schemes for solving the boundary value problem for the modified Helmholtz equation and generalized impedance boundary condition. The approaches are based on the reduction of the problem to the boundary integral equation with a hyper-singular kernel. In the first scheme the hyper-singular integral operator is treated by splitting off the singularity technique whereas in the second scheme the idea of numerical differentiation is employed. The solvability of the boundary integral equation and convergence of the first method are established. Exponential convergence for analytic data is exhibited by numerical examples. (C) 2021 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.Y. All rights reserved.

In this paper, we establish sufficient conditions for various stability aspects of a nonlinear Volterra integro-dynamic matrix Sylvester system on time scales. We convert the nonlinear Volterra integro-dynamic matrix Sylvester system on time scale to an equivalent nonlinear Volterra integro-dynamic system on time scale using vectorization operator. Sufficient conditions are obtained to this system for stability, asymptotic stability, exponential stability, and strong stability. The obtained results include various stability aspects of the matrix Sylvester systems in continuous and discrete models.

In this paper we propose and analyze an hp-version of the composite Gauss-Legendre quadrature rule which includes local subdivision of the integration domain and local variation of the number of quadrature nodes employed on each subinterval. We derive an a priori error estimate that is fully explicit in the local length of each subinterval, in the local number of quadrature nodes employed on each subinterval, and in the local regularity of the integrand. In particular, for integrands with singularities at one or both endpoints, we prove that exponential convergence can be achieved by using geometric partition of the integration domain and linearly increasing of the number of quadrature nodes in successive subintervals. Numerical experiments are provided to illustrate the theoretical results.