One-step Block Research Articles

An optimized hybrid block method for solving nonlinear singularly perturbed problems using non-uniform Jain-type meshes

This work aims to implement a one-step block method with three optimized hybrid points for solving singularly perturbed nonlinear initial value problems on non-uniform meshes. The basic properties of the method are investigated and it is shown that the proposed method is consistent, zero-stable, convergent and has a large region of absolute stability. The presented numerical results confirm the efficiency and accuracy of the proposed method.

One-step block scheme with optimal hybrid points for numerical integration of second-order ordinary differential equations

In this paper, a one-step block of optimized hybrid schemes for the numerical integration of second-order initial value problems (IVP) of ordinary differential equations (ODE) is constructed via collocation techniques. The developed scheme is obtained by considering two intra-step nodal points as hybrid points, which are chosen in order to achieve optimized errors of the main formulae approximating the solution such that 0 < v1 < v 2 < 1 where v1 and v2 are defined as hybrid points. The characteristics of the developed scheme are analyzed. Application of the new scheme on some second-order IVPs shows the accuracy and effectiveness of the scheme compared with some existing methods.

A variable step-size implementation of a new one-step block method for integrating Burgers' model equation

This article proposes a variable stepsize implementation of an improved one-step block method (OBM) that uses two intermediate points to solve Burgers' equation. The model's problems under consideration are semi-discretized by reducing them into an equivalent differential system using the method of lines. The OBM is improved with an embedding-like method executed in variable stepsize mode. A few of Burgers' real-life model problems are numerically solved to test the accuracy of our error estimation and mesh selection strategy. The numerical results confirmed that the variable stepsize implementation used in this manuscript performed better than the fixed stepsize implementation.

Janus Effect of Lewis Acid Enables One-Step Block Copolymerization of Ethylene Oxide and N-Sulfonyl Aziridine.

One-step sequence-selective block copolymerization requires stringent catalytic control of monomers relative activity and enchainment order. It has been especially rare for AnBm-type block copolymers from simple binary monomer mixtures. Here, ethylene oxide (EO) and N-sulfonyl aziridine (Az) compose a valid pair provided with a bicomponent metal-free catalyst. Optimal Lewis acid/base ratio allows the two monomers to strictly block-copolymerize in a reverse order (EO-first) as compared with the conventional anionic route (Az-first). Livingness of the copolymerization facilitates one-pot synthesis of multiblock copolymers by addition of mixed monomers in batches. Calculation results reveal that a Janus effect of Lewis acid on the two monomers is key to enlarge the activity difference and reverse the enchainment order.

A One-Step Block Hybrid Integrator for Solving Fifth Order Korteweg-de Vries Equations

Fifth-order Korteweg-de Vries (KdV) equations, arise in modeling waves phenomena such as the propagation of shallow water waves over a flat surface, gravity-capillary waves and sound waves in plasmas. In this work, a one-step block hybrid linear multistep method was derived using the collocation technique, to solve fifth-order KdV models via the Method of Line (MoL). The consistency, stability and convergence of the method were established. The efficiency of the method can be seen from comparison of the exact solutions of problems and other methods cited from literature.

The Direct Two-Point Block One-Step Method Which is efficient and Suitable for Solving Second-Order Differential Equation

Abstract: A direct two-point block one-step method for solving general second-order ordinary differential equations (ODEs) directly is presented in this paper. The one-step block method will solve the second-order ODEs without reducing to first-order equations. The direct solutions of the general second-order ODEs will be calculated at two points simultaneously using variable step size.Two point four step direct implicit block method is developed for solving directly the second order system of ordinary differential equations (ODEs) using variable step size. The method will estimate the solutions of Initial Value Problems (IVPs) at two points simultaneously by using four backward steps. The method developed is suitable for the numerical integration of non stiff and mildly stiff differential systems. Numerical results are given to compare the efficiency of the developed method to the existence block method.

One‐Step Synthesis of Lignin‐Based Triblock Copolymers as High‐Temperature and UV‐Blocking Thermoplastic Elastomers

This work utilizes frustrated Lewis pairs consisting of tethered bis-organophosphorus superbases and a bulky organoaluminum to furnish the highly efficient synthesis of well-defined triblock copolymers via one-step block copolymerization of lignin-based syringyl methacrylate and n-butyl acrylate, through di-initiation and compounded sequence control. The resulting thermoplastic elastomers (TPEs) exhibit microphase separation and much superior mechanical properties (elongation at break up to 2091 %, tensile strength up to 11.5 MPa, and elastic recovery up to 95 % after 10 cycles) to those of methyl methacrylate-based TPEs. More impressively, lignin-based tri-BCPs can maintain TPEs properties up to 180 °C, exhibit high transparency and nearly 100 % UV shield, suggesting potential applications in temperature-resistant and optical devices.

Optimization of One-Step Block Method for Solving Second-Order Fuzzy Initial Value Problems

In this article, we present a one-step hybrid block method for approximating the solutions of second-order fuzzy initial value problems. We prove the stability and convergence results of the method and present several examples to illustrate the efficiency and accuracy of the proposed method. The numerical results are compared with the existing ones in the literature.

One-Step Block Copolymer Templated Synthesis of Bismuth Oxybromide for Bisphenol A Degradation: An Extended Study from Photocatalysis to Chemical Oxidation

An energy and cost-effective method was reported for bismuth oxybromide (BiOBr) preparation, and its versatile applications were investigated from the well-established photocatalysis to chemical ox...

Numerical solution to Volterra Integro-Differential Equations of the second kind by hybrid one-step block method

The hybrid block one-step method of order four is presented and implemented to solve first order Volterra Integro-Differential Equations (VIDEs). The technique is developed using Lagrange interpolation method. The numerical solutions of VIDEs will be solved at two-point concurrently using the proposed numerical method. Properties of the method such as consistency, order and region of absolute stability are explored. After that, the linear and nonlinear of VIDEs has been solved and the numerical results show that the accuracy and efficiency of the hybrid block method is better than the other methods which is RK4 and VIM.

The Treatment of Second Order Ordinary Differential Equations Using Equidistant One-step Block Hybrid

A general one-step hybrid block method with equidistant of order 6 has been successfully developed for the direct solution of second order IVPs in this article. Numerical analysis shows that the developed method is consistent and zero-stable which implies its convergence. The analysis of the new method is examined on two highly and mildly stiff second-order initial value problems to illustrate the efficiency of the method. It is obvious that our method performs better than the existing method within which we compare our result with. Hence, the approach is an adequate one for solving special second order IVPs.

An efficient optimized adaptive step-size hybrid block method for integrating differential systems

This paper deals with the development, analysis and implementation of an optimized hybrid block method having different features, for integrating numerically initial value ordinary differential systems. The hybrid nature of the proposed one-step scheme allows us to bypass the first Dahlquist’s barrier on linear multi-step methods. The theory of interpolation and collocation has been used in the development of the method. We assume an appropriate polynomial representation of the theoretical solution of the problem and consider three off-step points in a one-step block. One of these three off-step points is fixed and the other two off-step points are optimized in order to minimize the local truncation errors of the main method and other additional formula. The resulting scheme is of order five having the property of A-stability. An embedded-type approach is used in order to formulate the proposed method in adaptive form, showing a high efficiency. The adaptive method is tested on well-known differential systems viz. the Robertson’s system, a Gear’s system, a system related with Jacobi elliptic functions, the Brusselator system, and the Van der Pol system, and compared with some well-known numerical codes in the scientific literature.

A one-step extended block hybrid formulae for solving orbital problems

A one-step extended block hybrid formulae for solving orbital problems

One-Step Block Copolymer Synthesis versus Sequential Monomer Addition: A Fundamental Study Reveals That One Methyl Group Makes a Difference

Block copolymers of polyisoprene and polystyrene are key materials for polymer nanostructures as well as for several commercially established thermoplastic elastomers. In a combined experimental and kinetic Monte Carlo simulation study, the direct (i.e., statistical) living anionic copolymerization of a mixture of isoprene (I) and 4-methylstyrene (4MS) in nonpolar media was investigated on a fundamental level. In situ 1H NMR spectroscopy enabled to directly monitor gradient formation during the copolymerization and to determine the nature of the gradient. In addition, a precise comparison with the established copolymerization of isoprene and styrene (I/S) was possible. Statistical copolymerization in both systems leads to tapered block copolymers due to an extremely slow crossover from isoprene to the styrenic monomer. For the system I/4MS the determination of the reactivity ratios shows highly disparate values with rI = 25.4 and r4MS = 0.007, resulting in a steep gradient of the comonomer composition. Th...

The Construction of Implicit One-step Block Hybrid Methods with Multiple Off-grid Points for the Solution of Stiff Differential Equations

The Construction of Implicit One-step Block Hybrid Methods with Multiple Off-grid Points for the Solution of Stiff Differential Equations

Implicit One-Step Block Hybrid Third-Derivative Method for the Direct Solution of Initial Value Problems of Second-Order Ordinary Differential Equations

A new one-step block method with generalized three hybrid points for solving initial value problems of second-order ordinary differential equations directly is proposed. In deriving this method, a power series approximate function is interpolated at {xn,xn+r} while its second and third derivatives are collocated at all points {xn,xn+r,xn+s,xn+t,xn+1} in the given interval. The proposed method is then tested on initial value problems of second-order ordinary differential equations solved by other methods previously. The numerical results confirm the superiority of the new method to the existing methods in terms of accuracy.

Generalized Hybrid One-Step Block Method Involving Fifth Derivative for Solving Fourth-Order Ordinary Differential Equation Directly

A general one-step three-hybrid (off-step) points block method is proposed for solving fourth-order initial value problems of ordinary differential equations directly. A power series approximate function is employed for deriving this method. The approximate function is interpolated atxn,xn+r,xn+s,xn+twhile its fourth and fifth derivatives are collocated at all pointsxi,i=0,r,s,t,1, in the interval of approximation. Several fourth-order initial value problems of ordinary differential equations are then solved to compare the performance of the proposed method with the derived methods. The analysis of the method reveals that the method is consistent and zero stable concluding that the method is also convergent. The numerical results demonstrate the superiority of the new method over the existing ones in terms of error.

Solving delay differential equations by the five-point one-step block method using Neville's interpolation

A five-point one-step block method based on the Newton backward divided difference formulae for the solution of first-order delay differential equations is derived. The proposed block method will approximate the solutions of initial value problems at five points simultaneously using variable step size. The approximation of the delay term is calculated using Neville's interpolation. The block method will be formulated in terms of linear multistep method, but the method is equivalent to one-step method. The order of the block method will be discussed. The P-stability and Q-stability regions of the block method using Neville's interpolation for the delay term are presented for a fixed step size. Numerical results are given to show the efficiency of the proposed method and compared with the existing method.

Solving second order delay differential equations by direct two and three point one-step block method

In this paper we present a two point and three point one-step block method for solving second order delay differential equations (DDEs). The one-step block method will solve directly the second order DDEs without reducing to first order equations. The two point and three point one-step block method will compute the solutions for the DDEs at two and three points simultaneously along the interval. These methods will solve the retarded type of DDE of single delay using constant step size. The P- stability and Q-stability are also discussed. The numerical results are presented to illustrate the performance of those block method for solving delay differential equations.

One-Step Poly(styrene-alt-maleic anhydride)-block-poly(styrene) Copolymers with Highly Alternating Styrene/Maleic Anhydride Sequences Are Possible by Nitroxide-Mediated Polymerization

Styrene/maleic anhydride (S/MA) copolymerizations at initial molar MA feed concentrations fMA,0 = 0.10−0.55 were done at 80−110 °C in 50 wt % 1,4-dioxane using 5 mol % {tert-butyl-[1-(diethoxyphosphoryl)-2,2-dimethylpropyl]amino}oxidanyl (SG1) relative to -2-({tert-butyl-[1-(diethoxyphosphoryl)-2,2-(dimethylpropyl)amino}oxy)-2-methylpropionic acid (BlocBuilder). As fMA,0 increased, the apparent rate constants increased dramatically. Copolymerizations performed with nearly equimolar feeds fMA,0 = 0.45−0.55 at 110 °C had apparent rate constants ∼20 times that of styrene at the same temperature. S/MA copolymerizations performed at 110 °C with practically equimolar feeds resulted in nearly alternating S/MA/S structures while those performed with fMA,0 = 0.1−0.4 possessed little to no S/MA/S alternating structures as indicated by 13C NMR spectroscopy. However, NMP of S/MA done at 90 °C with a styrene-rich feed composition (fMA,0 = 0.2) resulted in a poly(S-alt-MA)-block-poly(S) diblock in a single step. 13C NMR revealed two significant populations of triads corresponding to S/MA/S alternating sequences and to S/S/S homopoly(styrene) sequences. BlocBuilder enabled the polymerization of these one-step block copolymers with a strictly alternating S/MA first block by NMP, thus permitting NMP to join reversible addition−fragmentation transfer polymerization as another controlled radical polymerization method to attain alternating S/MA microstructures.