Interval Of Convergence Research Articles

A new modification of the Adomian decomposition method for solving boundary value problems for higher order nonlinear differential equations

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.

Application of Multi-Step Differential Transform Method on Flow of a Second-Grade Fluid over a Stretching or Shrinking Sheet

In this study, a reliable algorithm to develop approximate solutions for the problem of fluid flow over a stretching or shrinking sheet is proposed. It is depicted that the differential transform method (DTM) solutions are only valid for small values of the independent variable. The DTM solutions diverge for some differential equations that extremely have nonlinear behaviors or have boundary-conditions at infinity. For this reason the governing boundary-layer equations are solved by the Multi-step Differential Transform Method (MDTM). The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization, or perturbation. It is a semi analytical-numerical technique that formulizes Taylor series in a very different manner. By applying the MDTM the interval of convergence for the series solution is increased. The MDTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. It is predicted that the MDTM can be applied to a wide range of engineering applications.

A multi-step differential transform method and application to non-chaotic or chaotic systems

The differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka–Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multi-step DTM, classical DTM and the classical Runge–Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed.

On the convergence profile of a discretized scheme for a two-dimensional constrained optimal control problem

The convergence profile of a discretized schame for an optimal control problem constrained by ordinary differential equation with matrix coefficients is examined. Various penalty parameters are considered for the penalty function method. It is observed that convergence is exhibited for these penalty parameters after certain number of iterations with a predetermined interval of convergence. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 537-542

Solving nonlinear partial differential equations using the modified variational iteration Padé technique

In this paper, the modified variational iteration method (MVIM) is reintroduced with the enhancement of Padé approximants to lengthen the interval of convergence of VIM or MVIM when used alone in solving nonlinear problems. KdV, mKdV, Burger's and Lax's equations are used as examples to illustrate the effectiveness and convenience of the proposed technique.

Convergence and Formal Manipulation of Series from the Origins of Calculus to About 1730

In this paper I illustrate the evolution of series theory from Leibniz and Newton to the first decades of the eighteenth century. Although mathematicians used convergent series to solve geometric problems, they manipulated series by a mere extension of the rules valid for finite series, without considering convergence as a preliminary condition. Further, they conceived of a power series as a result of a process of the expansion of a finite analytical expression and thought that the link between series and analytical expression was not restricted to the interval of convergence. This gave rise to formal aspects that became progressively more evident while the complexity of the theory increased. Asymptotic and recurrence series emerged as the result of the natural evolution of theory, without the difference between these infinite processes with respect to ordinary series being highlighted. The various aspects of dealing with series can be viewed today as derived from different definitions of the sum. However, mathematicians of that time always considered them as different approaches to a single theory that was based on a unique notion of the sum (even though, later in the eighteenth century, they disagree about what this notion was).

Maclaurin Taylor Series for Transcendental Functions: A Graphing-Calculator View of Convergence

Most calculus students can perform the manipulation necessary for a polynomial approximation of a transcendental function. However, many do not understand the underlying concept. Graphing-calculator technology can be used to bridge this gap between the concept of an interval of convergence for a series and polynomial approximations. Calculus-reform textbooks usually treat this topic by displaying lower-order Maclaurin series approximations to selected transcendental functions to encourage discussions of intervals of convergence. Some textbooks display lower-order Taylor polynomials for ln x expanded about x = 1. This article presents a way to examine the topic in more depth.

Efficient Threshold Circuits for Power Series

We show that functions with convergent real power series can be well approximated by two classes of polynomial-size small-weight threshold circuits: depth-three circuits with threshold gates on all levels and depth-four circuits with threshold gates on the first two levels and AND–OR gates on the last two. This is done without restricting the input to a fixed closed subinterval of the interval of convergence of the series. We also point out that rational functions and the logarithm of x in base b can be well approximated by the same classes of circuits when both x and b are given as input.

Comparing software for interactive solution of systems of nonlinear algebraic equations

Four general purpose interactive numerical solution packages: MAPLE, MATLAB, MATHEMATICA and POLYMATH are compared with regard to their performance in solving systems of nonlinear algebraic equations typical to chemical engineering applications. Criteria for ease of learning, ease of use and user-friendliness of the packages are introduced. Benchmark problems are used to demonstrate that the selection of good initial estimates is still a critical issue in solving highly nonlinear problems with any of these programs. It is concluded that of the programs tested, the performance of MATHEMATICA and POLYMATH is the best for systems of NLEs, the interval of convergence of POLYMATH and MAPLE is the widest for a single NLE and POLYMATH possesses the most “user-friendly” features.

The convergence of energy expansions for molecules in electrostatic fields: A linear‐response coupled‐cluster study

The radii of convergence of power series expansions describing energy of a molecule in external electrostatic field are investigated usingD’Alembert ratio test, standard and generalized Cauchy–Hadamardcriteria, and Pade approximants. The corresponding coefficients at various field and field‐gradient components, representing multipole moments and (hyper)polarizabilities and including terms of tenth or even twentieth order, are determined using an ab initio linear responsecoupled‐cluster theory. Most calculations are performed for the HF molecule described by the basis set of double zeta quality, while the role of basis set is discussed by comparing the results with estimates of the radii of convergence obtained with the basis set of [5s3p2d/3s2p] quality. Emphasis is placed on the dependence of the interval of convergence of power series expansion describing energy of a molecule in applied electrostatic field on the nuclear geometry. The results might have important implications for various numerical methods used to calculate electrostatic molecular properties as functions of the internuclear geometry, including the finite‐field andfixed‐point‐charge approaches.

Power-series expansions as fitting functions of potential-energy curves

A comparative study of the most important power-series expansions, Dunham, Simons–Parr–Finland, Ogilvie–Tipping, Thakkar, Engelke, Mattera, Surkus and Huffaker, as fitting functions of potential-energy curves is reported. A study of the leading terms and intervals of convergence is also shown. As an example, the calculation of the interval of convergence for an Engelke's series is given. The method is applied to the molecules: CO (X 1Σ+), H2(X 1Σ+g) and 7LiH (X 1Σ+ and A 1Σ+). An analysis of the variation for the leading term of the power-series expansions with two non-linear parameters is presented for CO(X 1Σ+). The optimum non-linear parameters are obtained when the left-hand side of the interval of convergence is very near and below the first point of the input potential. Moreover, we observed that a good fit through the leading term of a power-series expansion is obtained for Engelke, Mattera or Surkus functions with two non-linear parameters. For fitting power-series expansions with an intermediate number of basis functions it is better to use a Thakkar or Huffaker type function with only one non-linear parameter.

Endpoint weak boundedness of some polynomial expansions

Let w( x) = (1 − x) α (1 + x) β on [− 1, 1], α,β⩾ − 1 2 , and for each function f let S n f be the nth expansion in the corresponding orthonormal polynomials. We show that the operators f → uS n ( u −1 f) are not of weak ( p, p)-type, where u is another Jacobi weight and p is an endpoint of the interval of mean convergence. The same result is shown for expansions associated to measures of the form dv = w( x) dx + Σ k i=1 M i δ a i .

Weighted Weak Behaviour of Fourier‐Jacobi Series

AbstractLet w(x) = (1 ‐ x)α (1 + x)β be a Jacobi weight on the interval [‐1, 1] and 1 < p < ∞. If either α > −1/2 or β > −1/2 and p is an endpoint of the interval of mean convergence of the associated Fourier‐Jacobi series, we show that the partial sum operators Sn are uniformly bounded from Lp,1 to Lp,∞, thus extending a previous result for the case that both α, β > −1/2. For α, β > −1/2, we study the weak and restricted weak (p, p)‐type of the weighted operators f→uSn(u−1f), where u is also Jacobi weight.

Calculus with Analytic Geometry.

Calculus with Analytic Geometry.

On an Abstract Nonlinear Cauchy-Kowalewski Theorem – a Variant of L. Nirenberg’s and T. Nishida’s Proof

The paper deals with a variant of the proof of an abstract nonlinear Cauchy-Kowalewski theorem. The original proof as well as its modification regarded in this paper make use of the method of successive approximations where in both cases the interval of convergence is characterized by an infinite product. In the present paper the infinite product of the original proof is replaced by one with arbitrary factors. In this way in some cases the convergence is proved in a larger interval.

The Interval of Convergence and Limiting Functions of a Hyperpower Sequence

The Interval of Convergence and Limiting Functions of a Hyperpower Sequence

The Interval of Convergence and Limiting Functions of a Hyperpower Sequence

The Interval of Convergence and Limiting Functions of a Hyperpower Sequence

Evolution of a Late Mesozoic back-arc fold and thrust belt, Northwestern Great Basin, U.S.A.

The Luning-Fencemaker fold and thrust belt was active from the Middle or Late Jurassic through the Early Cretaceous and involves rocks of the Mesozoic marine-province of the northwestern Great Basin. Rocks of the marine-province were deposited in a back-arc basin bound on the west by the Sierran arc. They are underlain by simatic crust formed either in a Paleozoic marginal-basin formed on the west coast of North America or in an exotic oceanic-arc accreted to North America in the Permo-Triassic. Deposition in the marine-province was localized by the underlying simatic crust, which later controlled the locus of back-arc thrusting. To the south, in areas of the back-arc region overlying continental crust, Mesozoic rocks are terrestrial volcanic and volcanogenic deposits. Rocks of this area are not disrupted by thrusts of significant magnitude until the southwestern end of the Sevier thrust belt is encountered. During contraction, the Luning-Fencemaker thrust belt underwent several hundred kilometers of NW-SE shortening which was not developed in the Sierra Nevada to the west. The arc and back-arc regions were separated by a regionally extensive left-lateral fault system that was active during back-arc thrusting. The marine-province was partially coupled with the Sierra Nevada during transpressional motion on the left-lateral fault, resulting in imbrication and emplacement of the Luning-Fencemaker thrust belt. In the continental region to the south, decoupling was more complete and only minor thrusting was associated with the southerly displacement of the Sierran arc. Farther south, where the Paleozoic miogeoclinal hingeline was encountered, conditions apparently changed and the Sevier thrust belt developed. Timing relations suggests that the Andean-type margin of the western United States had paired back-arc thrust belts (Luning-Fencemaker and Sevier) for much of the late Mesozoic. The belts were separated by a region that underwent relatively minor internal shortening. The intervening region probably was decoupled at depth and the thrust belts may have shared a décollement. Development of the Luning-Fencemaker thrust belt and the associated left-lateral fault system extended over a period of 40–50 m.y. and is best explained as a product of plate convergence. The NW-SE shortening axis for the thrust belt and the inferred left-slip on the strike-slip fault are not compatible with the generally accepted right-oblique subduction proposed for the Sierran arc since the Early Triassic. They suggest an interval of left-oblique convergence for the Middle to Late Jurassic and Early Cretaceous. Regionally developed structures indicate a return to right-oblique convergence in the late Early Cretaceous.

On a strong Tauberian result

We consider the determination of the behavior of a distribution function F at its endpoints in terms of the behavior of its Laplace-Stieltjes transform Ω at the limits of its interval of convergence. The results extend various known strong and weak results to a larger class of distributions via a relatively straightforward technique based on the weak convergence of suitably normalized associated distributions. An application and examples are considered briefly.

Tectonic implications of a late Mesozoic fold and thrust belt in northwestern Nevada

The Luning-Fencemaker fold and thrust belt of northwestern Nevada formed east of the Mesozoic Sierran arc in a back-arc setting where noncontinental crust localized marine deposition and subsequent deformation. The fold-thrust belt was active in Middle or Late Jurassic and Early Cretaceous time and evolved during major northwest-southeast contraction. Comparable shortening was not developed in the Sierran arc, and for much of the Mesozoic the arc and the back-arc regions were at least partially decoupled and separated by the regionally extensive Pine Nut fault system, which underwent substantial left-lateral displacement. Development of the fold-thrust belt and associated strike-slip fault is attributed to oblique subduction. The shortening axis of the fold-thrust belt and inferred displacement of the strike-slip fault are incompatible with right-oblique subduction and call for an interval of left-oblique convergence during part of Jurassic-Cretaceous time.