Weber argues that causal modelers face a dilemma when they attempt to model systems in which the underlying mechanism operates according to some set of differential equations. The first horn is that causal models of these systems leave out certain causal effects. The second horn is that causal models of these systems leave out time-dependent derivatives, and doing so distorts reality. Either way causal models of these systems leave something important out. I argue that Weber’s reasons for thinking (...) causal modeling is limited in this domain are lacking. (shrink)

Differential equations of second order appear in physical applications such as fluid dynamics, electromagnetism, acoustic vibrations, and quantum mechanics. In this paper, necessary and sufficient conditions are established of the solutions to second-order half-linear delay differential equations of the form ς y u ′ y a ′ + ∑ j = 1 m p j y u c j ϑ j y = 0 for y ≥ y 0, under the assumption ∫ ∞ ς η − (...) 1 / a d η = ∞. We consider two cases when a c j, where a and c j are the quotient of two positive odd integers. Two examples are given to show effectiveness and applicability of the result. (shrink)

There is a similarity between the most technical scientific reasoning and the most humanistic literary reasoning. While engineers and historians make use of both metaphors and stories, engineers specialize in metaphors, and historians in stories. Placing metaphor, or pure comparison, at one end of a scale and simply a listing of events, or pure story, at the other, it can be seen that what connects them is a theme. The theme providing the connecting link between poles for both the engineer (...) and the historian is time. Themes in engineering that mention time are those titled differential equations. The differential equation is story-like because it speaks of time and therefore organizes experience in time, at least implicitly. Time- speaking themes shape the raw experience, as a story does when it is more than a mere unthematized chronicle. The chaotic nature of non-linear differential equations parallels the chaotic nature of history in that large results need not have large causes. Narration becomes difficult in chaotic systems because the knowledge of initial conditions is rarely sufficiently detailed to allow for accurate explanation or prediction for either engineers or historians. (shrink)

In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures.Let ℜ be an expansion of the real field (ℝ, +, ·).A differentiable mapF= (F1,…,F1): (a, b) → ℝiisℜ-Pfaffianif there existsG: ℝ1+l→ ℝldefinable in ℜ such thatF′(t) =G(t, F(t)) for allt∈ (a, b) and each component functionGi: ℝ1+l→ ℝ is independent of the lastl−ivariables (i= 1, …,l). If ℜ is o-minimal andF: (a, b) → ℝlis ℜ-Pfaffian, then (ℜ,F) is o-minimal (Proposition (...) 7). We say thatF: ℝ → ℝlis ultimately ℜ-Pfaffian if there existsr∈ ℝ such that the restrictionF↾(r, ∞) is ℜ-Pfaffian. (In general,ultimatelyabbreviates “for all sufficiently large positive arguments”.)The structure ℜ isclosed under asymptotic integrationif for each ultimately non-zero unary (that is, ℝ → ℝ) functionfdefinable in ℜ there is an ultimately differentiable unary functiongdefinable in ℜ such that limt→+∞[g′(t)/f(t)] = 1- If ℜ is closed under asymptotic integration, then ℜ is o-minimal and definesex: ℝ → ℝ (Proposition 2).Note that the above definitions make sense for expansions of arbitrary ordered fields. (shrink)

Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind’s approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, (...) and a monograph on the Navier–Stokes equation to show structural methods like Dedekind’s have long been central to differential equations, and have philosophically respectable ontology and epistemology. (shrink)

We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby :445–486, 2009) and Crampin we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.

Euler's contributions to differential equations are so comprehensive and rigorous that any contemporary textbook on the subject can be regarded as a copy of Euler's Institutionum Calculi Integralis. Of course, Euler's work is an improvement of that of Leibniz, the Bernoullis, Newton and so many others before them, but still it's so outstanding that will be used in this paper as a reference to account for every previous or subsequent development in ODEs. Maybe Euler did not discovered (...) class='Hi'>differential equations, but he did not discovered less differential equations than Newton and Leibniz had discovered differential calculus a few decades before. (shrink)

Fractional calculus is nowadays an efficient tool in modelling many interesting nonlinear phenomena. This study investigates, in a novel way, the Ulam–Hyers and Ulam–Hyers–Rassias stability of differential equations with general conformable derivative. In our analysis, we employ some version of Banach fixed-point theory. In this way, we generalize several earlier interesting results. Two examples are given at the end to illustrate our results.

This essay will survey various considerations that arise when a branch of physics requires formulation in terms of partial differential equations (or some facsimile thereof). My examples will derive almost exclusively from classical continuum (=smeared out matter) mechanics. Although the relevant formal facts are well known, it is difficult to find coherent discussions of how the underlying phenomena ought to be viewed. In this paper, I will give an introduction to some of the issues, although I will confess (...) at the outset that I am not certain how these matters should be finally adjudicated. Certainly it would be helpful if philosophy paid more attention to the underlying issues.Some of the points I will discuss bear upon some of Professor Sklar’s observations. Although my attention will be largely directed towards the status of boundary conditions (and related forms of singular surface), many of my examples can be adapted, mutatis mutandis, to initial conditions as well. (shrink)

Diagrams have played an important role throughout the entire history of differential equations. Geometrical intuition, visual thinking, experimentation on diagrams, conceptions of algorithms and instruments to construct these diagrams, heuristic proofs based on diagrams, have interacted with the development of analytical abstract theories. We aim to analyze these interactions during the two centuries the classical theory of differential equations was developed. They are intimately connected to the difficulties faced in defining what the solution of a (...) class='Hi'>differential equation is and in describing the global behavior of such a solution. (shrink)

We present an age-structured model for erythropoiesis in which the mortality of mature cells is described empirically by a physiologically realistic probability distribution of survival times. Under some assumptions, the model can be transformed into a system of delay differential equations with both constant and distributed delays. The stability of the equilibrium of this system and possible Hopf bifurcations are described for a number of probability distributions. Physiological motivation and interpretation of our results are provided.

In this article, we make analysis of the implicit fractional differential equations involving integral boundary conditions associated with Stieltjes integral and its corresponding coupled system. We use some sufficient conditions to achieve the existence and uniqueness results for the given problems by applying the Banach contraction principle, Schaefer’s fixed point theorem, and Leray–Schauder result of the cone type. Moreover, we present different kinds of stability such as Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam–Rassias stability, and generalized Hyers–Ulam–Rassias stability by (...) using the classical technique of functional analysis. At the end, the results are verified with the help of examples. (shrink)

In this analysis, we develop a new approach to investigate the semianalytical solution of the delay differential equations. Mohand transform coupled with the homotopy perturbation method is called Mohand homotopy perturbation transform method and performs the solution results in the form of series. The beauty of this approach is that it does not need to compute the values of the Lagrange multiplier as in the variational iteration method, and also, there is no need to implement the convolution theorem (...) as in the Laplace transform. The main purpose of this scheme is to reduce the less computational work and the error analysis of the problems than others studied in the literature. Some illustrated examples are interpreted to confirm the accuracy of the newly developed scheme. (shrink)

An effective numerical scheme based on Euler wavelets is proposed for numerically solving a class of neutral delay differential equations. The technique explores the numerical solution via Euler wavelet truncated series generated by a set of functions and matrix inversion of some collocation points. Based on the operational matrix, the neutral delay differential equations are reduced to a system of algebraic equations, which is solved through a numerical algorithm. The effectiveness and efficiency of the technique (...) have been illustrated by several examples of neutral delay differential equations. The main advantages and key role of using the Euler wavelets in this work lie in the performance, accuracy, and computational cost of the proposed technique. (shrink)

To solve fractional delay differential equation systems, the Laguerre Wavelets Method is presented and coupled with the steps method in this article. Caputo fractional derivative is used in the proposed technique. The results show that the current procedure is accurate and reliable. Different nonlinear systems have been solved, and the results have been compared to the exact solution and different methods. Furthermore, it is clear from the figures that the LWM error converges quickly when compared to other approaches. When (...) compared with the exact solution to other approaches, it is clear that LWM is more accurate and gets closer to the exact solution faster. Moreover, on the basis of the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations. (shrink)

In this study, under some inequality conditions, necessary and sufficient conditions, using fixed-point theorem in cones, are established for the existence of C 1 -positive solutions for a class of second-order impulsive differential equations. Two examples are given in the last section to illustrate the abstract results.

Liquidity risk arises from the inability to unwind or hedge trading positions at the prevailing market prices. The risk of liquidity is a wide and complex topic as it depends on several factors and causes. While much has been written on the subject, there exists no clear-cut mathematical description of the phenomena and typical market risk modeling methods fail to identify the effect of illiquidity risk. In this paper, we do not propose a definitive one either, but we attempt to (...) derive novel mathematical algorithms for the dynamic modeling of trading volumes during the closeout period from the perspective of multiple-asset portfolio, as well as for financial entities with different subsidiary firms and multiple agents. The robust modeling techniques are based on the application of initial-value-problem differential equations technique for portfolio selection and risk management purposes. This paper provides some crucial parameters for the assessment of the trading volumes of multiple-asset portfolio during the closeout period, where the mathematical proofs for each theorem and corollary are provided. Based on the new developed econophysics theory, this paper presents for the first time a closed-form solution for key parameters for the estimation of trading volumes and liquidity risk, such as the unwinding constant, half-life, and mean lifetime and discusses how these novel parameters can be estimated and incorporated into the proposed techniques. The developed modeling algorithms are appealing in terms of theory and are promising for practical econophysics applications, particularly in developing dynamic and robust portfolio management algorithms in light of the 2007–2009 global financial crunch. In addition, they can be applied to artificial intelligence and machine learning for the policymaking process, reinforcement machine learning techniques for the Internet of Things data analytics, expert systems in finance, FinTech, and within big data ecosystems. (shrink)

Index sets are used to measure the complexity of properties associated with the differentiability of real functions and the existence of solutions to certain classic differential equations. The new notion of a locally computable real function is introduced and provides several examples of Σ04 complete sets.

A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter a, which is obtained using the collocation and least-squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in (...) this study. In five examples, the numerical results of the present method have been compared with other methods. For the first time, a -polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed. (shrink)

This article describes the emergence of formal methods in theory of partial differential equations in the French school of mathematics through Janet’s work in the period 1913–1930. In his thesis and in a series of articles published during this period, Janet introduced an original formal approach to deal with the solvability of the problem of initial conditions for finite linear PDE systems. His constructions implicitly used an interpretation of a monomial PDE system as a generating family of a (...) multiplicative set of monomials. He introduced an algorithmic method on multiplicative sets to compute compatibility conditions, and to study the problem of the existence and the uniqueness of a solution to a linear PDE system with given initial conditions. The compatibility conditions are formulated using a refinement of the division operation on monomials defined with respect to a partition of the set of variables into multiplicative and non-multiplicative variables. Janet was a pioneer in the development of these algorithmic methods, and the completion procedure that he introduced on polynomials was the first one in a long and rich series of works on completion methods which appeared independently throughout the twentieth-century in various algebraic contexts. (shrink)

In this paper, we consider the iterative algorithm for a boundary value problem of n -order fractional differential equation with mixed integral and multipoint boundary conditions. Using an iterative technique, we derive an existence result of the uniqueness of the positive solution, then construct the iterative scheme to approximate the positive solution of the equation, and further establish some numerical results on the estimation of the convergence rate and the approximation error.

In this study, we implemented a new numerical method known as the Chebyshev Pseudospectral method for solving nonlinear delay differential equations having fractional order. The fractional derivative is defined in Caputo manner. The proposed method is simple, effective, and straightforward as compared to other numerical techniques. To check the validity and accuracy of the proposed method, some illustrative examples are solved by using the present scenario. The obtained results have confirmed the greater accuracy than the modified Laguerre wavelet (...) method, the Chebyshev wavelet method, and the modified wavelet-based algorithm. Moreover, based on the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations. (shrink)

Solvency assessment is the core content of insurance supervision. In this paper, from the perspective of capital flow, the insurance company’s capital flow is regarded as a dynamic system, the stochastic differential equations model is established to describe its flow characteristics, and the existence of positive equilibrium point of the system is proved, as well as the conditions of stability at equilibrium point, that is, the requirements of the insurance company’s solvency. Furthermore, by using the numerical simulation method, (...) we get the strategy of insurance companies to deal with the solvency shortage when facing the change of external environment, and the strategy of insurance company to deal with solvency shortage is obtained. (shrink)

In this paper we study the connection between the use of evaluative language and the building of both personal and social identities, from the perspective of Dynamical System Theory . We primarily discuss two issues: 1) The use of evaluation (in the sense given to the term by Alba-Juez and Thompson (forthcoming)) as a means to the construction of both individual and group identities, thus exploring how the connection between linguistic choices and social identities is shaped by interactional needs for (...) stancetaking. In order to illustrate this connection, we examine examples of the use of evaluative language in a web social network, and we analyze some of the discourse elements showing ways of positioning that act as catalysts for the emergence of a multifactorial dynamic system of identities. 2) The consideration of Dynamical System Theory (DST) as a theoretical framework for the modeling of language and identity. Although originally a mathematical theory, DST has been adopted by cognitive science as a valid framework for the study of cognitive phenomena, on the grounds that natural cognition is a dynamical phenomenon. Within the realm of (socio) linguistics and pragmatics, this study is to a certain degree in line with some recent studies such as Gibbs (2010), Geeraerts, Kristiansen and Peirsman (2010), or Moreno Fernández (2012). Thus, we herein focus on how linguistic evaluation intervenes in the intricate dynamical system of identity, and even though we do not engage in complex mathematical disquisition, we argue that the idea and philosophical foundation underlying DST can lead us towards the ‘integration’ of the complex equation of identity construction, and that consequently the field has great potential for further research. (shrink)

We investigate the provability or nonprovability of certain ordinary mathematical theorems within certain weak subsystems of second order arithmetic. Specifically, we consider the Cauchy/Peano existence theorem for solutions of ordinary differential equations, in the context of the formal system RCA 0 whose principal axioms are ▵ 0 1 comprehension and Σ 0 1 induction. Our main result is that, over RCA 0 , the Cauchy/Peano Theorem is provably equivalent to weak Konig's lemma, i.e. the statement that every infinite (...) {0, 1}-tree has a path. We also show that, over RCA 0 , the Ascoli lemma is provably equivalent to arithmetical comprehension, as is Osgood's theorem on the existence of maximum solutions. At the end of the paper we digress to relate our results to degrees of unsolvability and to computable analysis. (shrink)

The demand of many scientific areas for the usage of fractional partial differential equations to explain their real-world systems has been broadly identified. The solutions may portray dynamical behaviors of various particles such as chemicals and cells. The desire of obtaining approximate solutions to treat these equations aims to overcome the mathematical complexity of modeling the relevant phenomena in nature. This research proposes a promising approximate-analytical scheme that is an accurate technique for solving a variety of noninteger (...) partial differential equations. The proposed strategy is based on approximating the derivative of fractional-order and reducing the problem to the corresponding partial differential equation. Afterwards, the approximating PDE is solved by using a separation-variables technique. The method can be simply applied to nonhomogeneous problems and is proficient to diminish the span of computational cost as well as achieving an approximate-analytical solution that is in excellent concurrence with the exact solution of the original problem. In addition and to demonstrate the efficiency of the method, it compares with two finite difference methods including a nonstandard finite difference method and standard finite difference technique, which are popular in the literature for solving engineering problems. (shrink)

The equivalence of Fermat's Last Theorem and the non-existence of solutions of a generalized n th order homogeneous hyperbolic partial differential equation in three dimensions and periodic boundary conditions defined in a cubic lattice is demonstrated for all positive integer, n > 2. For the case n = 2, choosing one variable as time, solutions are identified as either propagating or standing waves. Solutions are found to exist in the corresponding problem in two dimensions.

The dispute over the viability of various theories of relativistic, dissipative fluids is analyzed. The focus of the dispute is identified as the question of determining what it means for a theory to be applicable to a given type of physical system under given conditions. The idea of a physical theory's regime of propriety is introduced, in an attempt to clarify the issue, along with the construction of a formal model trying to make the idea precise. This construction involves a (...) novel generalization of the idea of a field on spacetime, as well as a novel method of approximating the solutions to partial-differential equations on relativistic spacetimes in a way that tries to account for the peculiar needs of the interface between the exact structures of mathematical physics and the inexact data of experimental physics in a relativistically invariant way. It is argued, on the basis of these constructions, that the idea of a regime of propriety plays a central role in attempts to understand the semantical relations between theoretical and experimental knowledge of the physical world in general, and in particular in attempts to explain what it may mean to claim that a physical theory models or represents a kind of physical system. This discussion necessitates an examination of the initial-value formulation of the partial-differential equations of mathematical physics, which suggests a natural set of conditions---by no means meant to be canonical or exhaustive---one may require a mathematical structure, in conjunction with a set of physical postulates, satisfy in order to count as a physical theory. Based on the novel approximating methods developed for solving partial-differential equations on a relativistic spacetime by finite-difference methods, a technical result concerning a peculiar form of theoretical under-determination is proved, along with a technical result purporting to demonstrate a necessary condition for the self-consistency of a physical theory. (shrink)

We investigate systems of ordinary differential equations with a parameter. We show that under suitable assumptions on the systems the solutions are computable in the sense of recursive analysis. As an application we give a complete characterization of the recursively enumerable sets using Fourier coefficients of recursive analytic functions that are generated by differential equations and elementary operations.

This paper is to investigate the existence and uniqueness of solutions for an integral boundary value problem of new fractional differential equations with a sign-changed parameter in Banach spaces. The main used approach is a recent fixed point theorem of increasing Ψ − h, r -concave operators defined on ordered sets. In addition, we can present a monotone iterative scheme to approximate the unique solution. In the end, two simple examples are given to illustrate our main results.