Interval Of Convergence Research Articles

On a New Two-Point Taylor Expansion With Applications

A new two-point Taylor series expansion is proposed. The expansion is slightly different than the classical definition. The coefficients are calculated as recursive relations in a general form. The two-point Taylor expansion is applied to several functions which are odd, even, neither odd nor even. Functions having finite interval of convergence or infinite interval of convergence are investigated. The conditions for convergence are derived and the results are compared with the results of single-point Taylor expansions as well as two-point Taylor expansions reported in the literature. It is found that for a finite radius of convergence, two-point Taylor expansions can have a single convergence interval as well as two separate convergence intervals. Generally speaking, two-point Taylor expansions better represent the real function when the series is truncated. The new two-point expansion and the classical two-point expansion produced identical results for all the problems treated. Based on the results of this analysis, the asymmetric two-point Taylor expansion presented here does not have an advantage compared to the classical symmetric expansion. An application of the series to solution of a variable coefficient differential equation is also treated.

On a New Two Point Taylor Expansion With Applications

A new two point Taylor series expansion is proposed. The expansion is slightly different than the classical definition. The coefficients are calculated as recursive relations in a general form. The two point Taylor expansion is applied to expressing two different functions, one of which has a finite interval of convergence and the other infinite interval of convergence. The conditions for convergence are derived and the results are compared with the results of single point Taylor expansions as well as two point Taylor expansions reported in the literature. It is found that for a finite radius of convergence, two point Taylor expansions can have a single convergence interval as well as two separate convergence intervals. Results of the new expansion are compared with the single point Taylor expansions as well as the classical two point Taylor expansion. Generally speaking two point Taylor expansions better represent the real function when the series is truncated. The new two point expansion and the classical two point expansion produced identical results for the problems treated. An application of the series to solution of a variable coefficient differential equation is also treated.

A robust and efficient hybrid solver for crystal plasticity

Conventional crystal plasticity (CP) solvers are based on a Newton-Raphson (NR) approach which use an initial guess for the free variables (often stress) to be solved. These solvers are limited by a finite interval of convergence and often fail when the free variable falls outside this interval. Solution failure results in the reduction of the time increment to be solved, thus convergence of the CP solver is a bottleneck which determines the computational cost of the simulation. The numerical stability of the slip law in its inverted form offers a solver that isn't vulnerable to poor pre-conditioning (initial guess) and can be used to progress to a solution from a stable starting point (i.e., from zero slip rate γ˙pk=0 s−1). In this paper, a novel formulation that enables the application of the slip law in its inverted form is introduced; this treats all slip systems as independent by approximating the Jacobian as a diagonal matrix, thus overcomes ill-defined and singular Jacobians associated with previous approaches. This scheme was demonstrated to offer superior robustness and convergence rate for a case with a single slip system, however the convergence rate for extreme cases with several active slip systems was relatively poor. Here, we introduce a novel ‘hybrid scheme’ that first uses the reverse scheme for the first stage of the solution, and then transitions to the forward scheme to complete the solution at a higher convergence rate. Several examples are given for pointwise calculations, followed by CPFEM simulations for FCC copper and HCP Zircaloy-4, which demonstrated solver performance in practise. The performance of simulations using the hybrid scheme was shown to require six to nine times fewer increments compared to the conventional forward scheme solver based on a free variable of stress and initial guess based on a fully elastic increment.

Generating Functions Related to the Fibonacci Substitution

In this paper, two generating function representations of the Fibonacci Substitution Tiling are derived and proven to converge on the interval -1<x<1. A sequence of signs for the Fibonacci Substitution is established along with a conjecture that the interval of convergence has an infinite number of zeroes.

On Some Fixed Point Iterative Schemes with Strong Convergence and Their Applications

In this paper, a new one-parameter class of fixed point iterative method is proposed to approximate the fixed points of contractive type mappings. The presence of an arbitrary parameter in the proposed family increases its interval of convergence. Further, we also propose new two-step and three-step fixed point iterative schemes. We also discuss the stability, strong convergence and fastness of the proposed methods. Furthermore, numerical experiments are performed to check the applicability of the new methods, and these have been compared with well-known similar existing methods in the literature.

Solutions of the Van der Pol Equation

Summary Presented in this paper are various solutions to the Van der Pol equation. Numerical solutions are utilized as an independent means of validating the various solutions discussed. A new solution in the form of a power series has been found. Although this solution is exact, its interval of convergence can only be estimated for a special case. Numerical experiments reveal that the power series solution can provide an exact solution over intervals where other approximate solutions are not valid. Thus, the new solution represents an additional solution that can complement other existing solutions. This work also emphasizes the importance and the role of computation. Although the power series solution along with the other approximate solutions mentioned are of theoretical interest, their restrictions limit their usefulness in real applications, and therefore numerical methods should also be considered.

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

Differential-algebraic equations (DAEs) are important tools to model complex problems in various application fields easily. Those DAEs with an index-3, even the linear ones, are known to cause problems when solving them numerically. The present article proposes a new algorithm together with its multistage form to efficiently solve a class of nonlinear implicit Hessenberg index-3 DAEs. This algorithm is based on the idea of applying the differential transform method (DTM) directly to the DAE without applying the traditional index reduction methods, which can be complex and often result in violations of the DAE constraints. Also, to deal with the nonlinear terms in the DAE, we approximate them using the Adomian polynomials. This new idea has given us a simple and efficient algorithm, which involves the solution of linear algebraic systems except for the initial recursion terms. This algorithm is easy to implement in Maple or Mathematica. Furthermore, to enlarge the interval of convergence of the power series solution obtained from the DTM, an algorithm for the multistage DTM is also given. Both algorithms are applied to solve two examples of highly nonlinear implicit index-3 Hessenberg DAEs. Numerical results show that the DTM can determine the exact solution in convergent power series, while the multistage DTM can compute accurate numerical solutions over large intervals.

РОЗВИТОК ФІНТЕХУ В УКРАЇНІ ТА ЙОГО ПРОГНОЗУВАННЯ НА ПРИКЛАДІ ІНДУСТРІЇ ПЛАТІЖНИХ ПОСЛУГ

The article examines the prerequisites for accelerating the development of fintech in Ukraine and in the world. Based on the definition of a fintech company as an organization working in the field of financial technology and using new business process models, it is reasoned that the value it offers to consumers is based on the use of information technology that has emerged during the active spread of the Internet. Ukrainian fintech market analysis demonstrates that there are no global companies in the Ukrainian domestic market. This is the result of its level of development and weak integration of Ukraine into the global financial market, as well as corruption, the uncertainty of the industry status, and relative poverty, which makes our country unprofitable for investment and infrastructure deployment of foreign fintech companies. Particular attention is paid to the market of payment services as a priority area of fintech companies. The structure of the payment services market in terms of the participating companies is described and its competitive parameters are determined. Despite the formal monopoly, in practice the market for services is competitive, industry leaders do not pursue aggressive competition policies. A comparative analysis of the fintech services and banking services markets dynamics in the field of payments is conducted by comparing the financial results of activities. The fintech companies’ gross aggregate profit growth forecast and their main competitor among banks in the field of payments are constructed. Based on optimistic, pessimistic, and realistic options, the time interval of income convergence of leading competitors in the payment services market is determined. The preconditions for changing the competitive situation in the field of payments are determined. The conducted analysis demonstrates that while maintaining the current trends, fintech companies as a generalized force will be able to become a powerful player in the payments market no earlier than 2028. Banks have up to ten years to adapt and adopt new rules of the game. Competition with fintech companies requires significant adjustments to banks' business strategies.

Characteristics of Modified Diffusion Analysis of Squeezed Hydro-magnetic Nanofluid Flow

This communication based on nanofluid theory is exhibited to study the squeezing mechanism. Hydro-magnetic phenomenon is incorporated in the analysis of nanofluid flow. The flow of heat and mass transport is interpreted by the constitutive relationship of modified diffusion model which overcomes the deficiencies in Fourier’s and Fick’s classical theories. The nano-fluid theory also incorporates the features of two important slip mechanisms namely Brownian diffusion and thermophoresis. Resulting heat and mass transport problems are figure out to develop the series solutions. Interval of convergence is established for computed series solutions. Behavior of emerging parameters in flow distributions are graphed and discussed. The major founds in this attempt is that higher Brownian diffusion strengthens the both heat and mass fluxes, while larger thermophoresis parameter enhances the heat flux but it weakens the mass flux. Moreover, both heat and mass flow field decays for enlarge estimation of thermal and solutal time relaxation parameters, respectively. Velocity field shows dual behavior for dominant magnetic parameter.

An Approximate Solution to Predator-prey Models Using The Differential Transform Method and Multi-step Differential Transform Method, in Comparison with Results of The Classical Runge-kutta Method

Predator-prey models are the building blocks of the ecosystems as biomasses are grown out of their resource masses. Different relationships exist between these models as different interacting species compete, metamorphosis occurs and migrate strategically aiming for resources to sustain their struggle to exist. To numerically investigate these assumptions, ordinary differential equations are formulated, and a variety of methods are used to obtain and compare approximate solutions against exact solutions, although most numerical methods often require heavy computations that are time-consuming. In this paper, the traditional differential transform (DTM) method is implemented to obtain a numerical approximate solution to prey-predator models. The solution obtained with DTM is convergent locally within a small domain. The multi-step differential transform method (MSDTM) is a technique that improves DTM in the sense that it increases its interval of convergence of the series expansion. One predator-one prey and two-predator-one prey models are considered with a quadratic term which signifies other food sources for its feeding. The result obtained numerically and graphically shows point DTM diverges. The advantage of the new algorithm is that the obtained series solution converges for wide time regions and the solutions obtained from DTM and MSDTM are compared with solutions obtained using the classical Runge-Kutta method of order four. The results demonstrated is that MSDTM computes fast, is reliable and gives good results compared to the solutions obtained using the classical Runge-Kutta method.

Approximate Analytical Solution of a Power-Law Pseudoplastic Fluid in a Boundary Layer over a Porous Plate: A New Application of Multi-Stage Parker-Sochacki Method

In this paper, based on Parker-Sochacki method for solving a system of differential equations,a multistage technique is developed for solving the nonlinear boundary layer equations of powerlawfluid on infinite domain. The problem domain is split into subintervals over which the boundaryvalue problem is replaced with a sequence of subproblems. In a shooting-like approach, the boundarycondition at infinity is converted to an equivalent initial condition. By recasting the problem as apolynomial system of first-order autonomous equations, the sub-problems are solved with Parker-Sochacki method with very high accuracy. The interval of convergence of the solution is deriveda-priorly in terms of the parameters of the polynomial system, which guides optimal choice of thediscretization parameter. The technique yielded a convergent piecewise continuous solution over theproblem domain. The results obtained, demonstrated graphically and in tables, compared well withexisting ones in the literature.

Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott

A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the ℏ u , v -curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.

Q-homotopy analysis method for solving nonlinear Fredholm integral equation of the second kind

Several scientific and engineering applications are usually described as integral equations. We propose a numerical approach for solving the type of Fredholm nonlinear boundary value problems in a finite domain. The paper aims to use the q-homotopy analysis method to estimate the solution to test the efficiency of the proposed method. Comparison with updated work is exacted. The obtained results show that the proposed method is very effective and convenient for nonlinear Fredholm integral equations. The interval of convergence of homotopy analysis method, if exists, is increased when using q-homotopy analysis method is more to converge. The result reveals that the q- homotopy analysis method is considered a good method for solving NFIES.

Using Padé Approximant Method to Solve the Mathematical Model of Tumor-Immune Interactions.

A mathematical model, in the form of a system of nonlinear ordinary differential equations, that describes the interaction between tumor cells and effective immune cells is proposed. An exact solution cannot be found to this system like many other nonlinear systems. Yet, approximate analytical solution is explored. This solution should have a large interval of convergence to be acceptable because the interaction can take many days to reach its steady state. Power series method is used to obtain a series solution. In this process, some auxiliary variables are used to transform the system of equations to polynomial form. However, this solution has a small radius of convergence, therefore, Padé approximant method is used to extend the domain of convergence. Hence, the obtained approximate analytical solution is valid over a large interval and has a remarkable accuracy when compared with numerical solution.

Limit theorems for some classes of power series type distributions

AbstractSeveral classes of distributions of power series type with finite and infinite radii of convergence are considered. For such distributions local limit theorems are obtained as the parameter of distribution tends to the right end of the interval of convergence. For the case when the convergence radius equals to 1, we prove an integral limit theorem on the convergence of distributions of random variables (1 − x)ξxas x → 1− to the gamma-distribution (ξx is a random variable with corresponding distribution of the power series type). The proofs are based on the steepest descent method.

Third grade fluid flow of stretching cylinder with heat source/sink

In this research, the impact of heat transfer on mixed convection steady third-grade fluid flow over an impermeable stretching cylinder with heat source is scrutinized. The investigation of mixed convection with non-Newtonian fluid is significant in geophysical and engineering fields. Appropriate transformations are alleged to obtain ordinary differential equations, which are later computed by using an analytical approach called the homotopy analysis methodology (HAM), and the interval of convergence is computed. Local Nusselt number and coefficient of skin friction values are computed numerically for novel parameters.

Maximum Interval of Stability and Convergence of Solution of a Forced Mathieu’s Equation

This paper investigates the maximum interval of stability and convergence of solution of a forced Mathieu’s equation, using a combination of Frobenius method and Eigenvalue approach. The results indicated that the equilibrium point was found to be unstable and maximum bounds were found on the derivative of the restoring force showing sharp condition for the existence of periodic solution. Furthermore, the solution to Mathieu’s equation converges which extends and improves some results in literature.

Fractional dynamics of an erbium-doped fiber laser model

In this paper we investigate the model of the time-fractional dynamics of an erbium-doped fiber laser model (TFDEFL) with Liouville–Caputo (LC), Caputo–Fabrizio–Caputo (CFC) and Atangana–Baleanu–Caputo (ABC) time-fractional derivatives. We employ the homotopy analysis transform method (HATM) to calculate approximate solutions for the TFDEFL model. This method gives the solution in the form of a rapidly convergent series that can ensure the convergence in solving the resultant series. We study the convergence analysis of HATM by computing the interval of convergence through the h-curves, the residual error function and the average residual error, respectively. We also show the effectiveness and accuracy of this method by comparing the approximate solutions based upon the LC, CFC and ABC time-fractional derivatives.

Three-dimensional Cartesian parabolic equation model with higher-order cross-terms using operator splitting, rational filtering, and split-step Padé algorithm.

An approximate form of three-dimensional Cartesian split-step marching solution for the acoustic parabolic equation is derived in order to obtain the efficient algorithm for sound propagation in the three-dimensional ocean. The operator splitting method is used to split the full exponential operator into three exponential operators for depth, cross-range, and the combination of the two. The first two terms are implemented with the split-step Padé algorithm and the final term is implemented with the Taylor series expansion in depth and cross-range operator. In order to resolve the divergence of Taylor approximation out of the interval of convergence, the rational filter of rectangular type is applied to the depth and cross-range operator. The use of the filter improves the stability of the solution but requires extra numerical burdens. Numerical issues involving the accuracy, efficiency, and stability of the proposed model are discussed and illustrated in an ocean wedge environment.

An optimal iterative algorithm for solving Bratu-type problems

Very Recently, Das et al. (J Math Chem 54:527–551, 2016) proposed a method based on variational iteration method for solving Bratu-type problems. In this work, we design a fast iterative scheme based on the optimal homotopy analysis method to obtain series solution for such problem. The proposed method contains a convergence control parameter which adjusts the interval of convergence of the series solution. This parameter is computed by minimizing the sum of the squared residual of the governing equation and with the help of which we obtain an optimal series solution to the problem. Further, our method avoids the usual time-consuming procedure to find the root of a nonlinear algebraic or transcendental equation for the undetermined coefficients. Besides the design of the method, convergence analysis and error estimate of the proposed method are supplemented. Two Bratu type problems are solved to demonstrate the efficiency and accuracy of the proposed method. Numerical results reveal the superiority of the proposed iterative scheme over a newly developed iterative method based on the variational iteration method (Das et al. 2016).