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In this work, we present an efficient semi-analytical method based on the Taylor series for solving nonlinear Volterra integro-differential equations, namely the differential transform method (DTM). The DTM provides a recursive relation for the coefficients of the Taylor series that is derived from the given equations. We provide a new recursive relation for the nonlinear Volterra integro-differential equations with complex nonlinear kernels. Since the DTM is based on the Taylor series, it is difficult to obtain accurate approximate solutions in a large domain. To overcome this difficulty, the standard DTM is applied in each subdomain, called the multistage differential transform method (MsDTM). We also present an convergence analysis for the proposed method. To demonstrate the efficiency of the proposed method, several numerical examples are performed and support the results in our analysis. [ABSTRACT FROM AUTHOR]

Highlights: [•] Use the real Lie algebra to generate a soliton hierarchy. [•] Generate a bi-Hamiltonian structure by the trace identity. [•] Obtain a hereditary recursion operator. [ABSTRACT FROM AUTHOR]

Abstract: Based on four sets of Lenard recursion sequences and zero-curvature equation, we derive the full positive flows of the Manakov hierarchy associated with a matrix spectral problem, from which some new nonlinear evolution equations are proposed. With the help of the Darboux transformation, soliton solutions of two new nonlinear evolution equations in the Manakov hierarchy are constructed. As two special reductions, the full positive flows of the coupled modified Korteweg–de Vries hierarchy and the Sasa–Satsuma hierarchy are deduced, in which some new nonlinear evolution equations are included. And then, we construct the Hamiltonian structures of the Manakov hierarchy and infinite conservation laws of several nonlinear evolution equations. [Copyright &y& Elsevier]

Abstract: In this work, we develop a variety of (2+1)-dimensional Burgers equations BE(N) of distinct orders. We use the recursion operator of the Burgers equation to derive these equations. The obtained equations possess multiple kink solutions the same as the multiple kink solutions of the Burgers hierarchy, but differ only in the dispersion relations. We finally establish a generalization for the dispersion relations. [Copyright &y& Elsevier]

Abstract: Integrability of polynomial differential system for any rational resonance ratio is studied. A new linear recursion formula for computation of generalized focus quantities and certain relation between two quantities from different definitions are given. As the applications, we discuss the generalized center conditions for two cubic Kolmogorov systems which are considered for the first time. [Copyright &y& Elsevier]

Abstract: In this work we present a new variety of (2+1)-dimensional mKdV (N) equations with distinct even orders. We use the recursion operator of the mKdV equation to develop achieve this goal. We show that these equations give multiple soliton solutions and multiple singular soliton solutions the same as the multiple soliton and multiple singular soliton solutions of the mKdV hierarchy except for the dispersion relations. We also develop a generalization for the dispersion relations and the order of these equations. [Copyright &y& Elsevier]

Abstract: In this paper we propose a new modified recursion scheme for the resolution of multi-order and multi-point boundary value problems for nonlinear ordinary and partial differential equations by the Adomian decomposition method (ADM). Our new approach, including Duan’s convergence parameter, provides a significant computational advantage by allowing for the acceleration of convergence and expansion of the interval of convergence during calculations of the solution components for nonlinear boundary value problems, in particular for such cases when one of the boundary points lies outside the interval of convergence of the usual decomposition series. We utilize the boundary conditions to derive an integral equation before establishing the recursion scheme for the solution components. Thus we can derive a modified recursion scheme without any undetermined coefficients when computing successive solution components, whereas several prior recursion schemes have done so. This modification also avoids solving a sequence of nonlinear algebraic equations for the undetermined coefficients fraught with multiple roots, which is required to complete calculation of the solution by several prior modified recursion schemes using the ADM. [Copyright &y& Elsevier]

Abstract: It is proved under appropriate assumptions that the solutions of a linear system of nonautonomous delay difference equations have finite limit at infinity. The results are based on a transformation of the delay difference system into a first-step recursion, where the companion matrices are well treatable from our point of view. Our theory is illustrated by examples, including a class of linear delay difference equations with unbounded coefficients. [ABSTRACT FROM AUTHOR]

Abstract: Integrable coupling with six potentials is first proposed by coupling a given 3×3 discrete matrix spectral problem. It is shown that coupled system of integrable equations can possess zero curvature representations and recursion operators, which yield infinitely many commuting symmetries. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebras, the Hamiltonian form is deduced for the lattice equations in the resulting hierarchy. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian system. [Copyright &y& Elsevier]

Abstract: We turn the physical Church–Turing Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and well-defined scientific problem(s): Examining recent controversies and causes for misunderstanding concerning the state of the Church–Turing Hypothesis (CTH), it is suggested to study the CTH ‘sharpened’ relative to an arbitrary but specific physical theory – rather than vaguely referring to “nature” in general. For this purpose we apply, and emphasize the utility of, concepts from philosophy: physical structuralism, ontological commitment, and constructivism. This general approach is then illustrated with some exemplary results on computability and complexity theory in computational physics. [Copyright &y& Elsevier]

Abstract: Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory and Algorithmic Statistics are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory – and its applied branches, say computational complexity theory or algorithmic information theory – and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general – but concise – analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory (The qualification ‘recursive’ here has nothing to do with ‘recursion theory’. Instead, this is a reference to the mathematical formalizations of the rational economic agent’s intertemporal optimization problems, in terms of Markov Decision Processes, (Kalman) Filtering and Dynamic Programming, where a kind of ‘recursion’ is invoked in the solution methods. The metaphor of the rational economic agent as a ‘signal processor’ underpins the recursive macroeconomic paradigm.), are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics. [Copyright &y& Elsevier]

Abstract: Using some contiguous function relations for the classical Gauss hypergeometric series , we establish a number of new recursion formulas for the Appell functionsandwhereSome important applications of our results to the evaluation of radiation field integrals are discussed and a listing of some useful reduction (and transformation) formulas is provided. [Copyright &y& Elsevier]

Abstract: The main purpose of the current paper is to prove that every positive solution of the delay difference equationwhere A, B ∈(0,∞), the initial conditions x −3k+1, x −3k+2,…, x 0 ∈(0,∞) and k ∈{1,2,3,…}, converges eventually to a period-k solution. We also give the results of computational examples to support our theoretical discussion. [Copyright &y& Elsevier]

Abstract: We elaborate on the following result which appears in that paper: if is consistent, then so is ZFC+[P = NP]. [Copyright &y& Elsevier]

Abstract: A procedure is given for constructing the exact integrals involving B-splines using recursion relations. The recursive integrals form the basic constituents for the exact evaluation of integrals that appear in the calculation of atomic and molecular properties. The method can be applied to solve differential equations as well as to produce a complete set of basis functions that may approximate a function arbitrarily well depending on the degree k and the number of B-splines that are employed in the approximation. The advantage of this method is that the recursions developed over a fixed interval represent exact results for the particular integral involved. Several examples are also provided to show how these recursion relations can be applied to evaluate integrals involving multiple B-splines of same or different degree. Closed forms of these integrals involve complicated recursion relations. [Copyright &y& Elsevier]

The symplectic-Hamiltonian formulation and recursion operator of the fifth-order Mikhailov–Novikov–Wang system are given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we give a recursion formula for the characteristic polynomial of Hessenberg matrices. As an application a new efficient computational algorithm for the determinant of a general Hessenberg matrix is presented. [Copyright &y& Elsevier]