Special Differential Equations Research Articles

Multi-step hybrid methods for special second-order differential equations y ″(t) = f(t,y(t))

In this paper, multi-step hybrid methods for solving special second-order differential equations yź(t) = f(t,y(t)) are presented and studied. The new methods inherit the frameworks of RKN methods and linear multi-step methods and include two-step hybrid methods proposed by Coleman (IMA J. Numer. Anal. 23, 197---220, 8) as special cases. The order conditions of the methods were derived by using the SN-series defined on the set SNT of SN-trees. Based on the order conditions, we construct two explicit four-step hybrid methods, which are convergent of order six and seven, respectively. Numerical results show that our new methods are more efficient in comparison with the well-known high quality methods proposed in the scientific literature.

A Four-Stage Fifth-Order Trigonometrically Fitted Semi-Implicit Hybrid Method for Solving Second-Order Delay Differential Equations

We derived a two-step, four-stage, and fifth-order semi-implicit hybrid method which can be used for solving special second-order ordinary differential equations. The method is then trigonometrically fitted so that it is suitable for solving problems which are oscillatory in nature. The methods are then used for solving oscillatory delay differential equations. Numerical results clearly show the efficiency of the new method when compared to the existing explicit and implicit methods in the scientific literature.

Extending high order derivatives for special differential equations of the form \(y' = f(y)\) by using monotonically labeled rooted trees.

This paper presents a review of the role played by labeled rooted trees to obtain derivatives for numerical solution of initial value problems in special case \(y' = f(y), y(x_0) = y_0\). We extend a process to find successive derivatives according to monotonically labeled rooted trees, and prove some relevant lemmas and theorems. In this regard, the  derivatives, of the monotonically labeled rooted trees with n vertices are presented by using the monotonically labeled rooted trees with k + n vertices. Eventually, this process is applied to trees without labeling.

INTEGRATION FOR SPECIAL THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS USING IMPROVED RUNGE-KUTTA DIRECT METHOD

In this paper, we derive an explicit four stage fifth-order Improved Runge-Kutta (IRKD) methodfor numerical integration of special third-order ordinary differential equation. The method proposed here is two-stepin nature and require less number of stages per step compared with the existing Runge-Kutta (RK) method. Thestability polynomial of the IRKD method is presented. Numerical results are given to illustrate the efficiency of theproposed method compared to the RK method and direct Runge-Kutta (RKD) method for solving special third-orderordinary differential equations. Keywords: Special third-order ordinary differential equations, Runge-Kutta method, IRKD method, StabilityPloynomial

Local stability and Hopf bifurcation of two-dimensional nonlinear descriptor system

Descriptor systems, which are also called differential-algebraic systems, singular systems, degenerate systems, constrained systems and so on, have been one of the major research topics in engineering field. However, problems related to the local stability and the Hopf bifurcation of nonlinear descriptor systems with time delay have not been thoroughly investigated. In this paper, we consider the dynamical behavior of two-dimensional nonlinear descriptor systems with time delay. First, local stability of the system is analyzed using the location of the roots of the characteristic equation of the corresponding linearized system. It is well known that the equilibrium of the linearized system is locally asymptotically stable if all roots of the corresponding characteristic equation locate in the left half of the complex plane, and they are uniformly bounded away from the imaginary axis. Otherwise, the equilibrium is unstable. The conditions for the existence of Hopf bifurcation are investigated in detail by using the time delay as a bifurcation parameter. Furthermore, based on the descriptor and the “neutral-type” model transformation, some special neutral differential equations can be equivalently transformed into the nonlinear descriptor systems. Finally, the correctness and the effectiveness of the theoretical analysis are justified by numerical examples.

Improvement and applications of secure outsourcing of scientific computations

Methods for solving equations play an important role in scientific computation and cryptography. Usually, many cryptographic protocols and scientific computation problems can be reduced to some differential equation or linear equation systems. However, the above mentioned scientific computation needs very much computation resource. Hence, super computer technology such as grid technology, cloud computation is slow to emerge. Cloud computation has strong computation power and helps users to deals with this very difficult computation. For security, we need improve some conditional computation methods for numerical or scientific outsourcing computation. In this paper some special algebraic and differential equations are studied. And some new practical verifiably secure outsourcing protocols are designed, which are better than the previous methods in the following terms. First, they can protect more secret parameters for abstract equations. Second, based on previous methods, random function can be chosen in the different ways and hence better security and privacy is obtained. Third, for nonlinear equations, we find more applications and new methods. These tools help us to find more methods to deal with more linear or nonlinear equations. At last, some verifiably secure outsourcing computation for integral computation is designed. These results help to outsource the computation for solving differential equations.

Two-step-by-two-step PIRKN-type PC methods based on Gauss–Legendre collocation points for nonstiff IVPs

This paper investigates parallel predictor–corrector (PC) iteration methods for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations of the form \(\mathbf{y}''(t)=\mathbf{f}(t,\mathbf{y}(t))\). The predictor methods are based on Adams-type formulas. The corrector methods are constructed by using coefficients of \(s\)-stage collocation Gauss–Legendre Runge–Kutta–Nystrom (RKN) methods based on \(\hat{c}_1,\dots ,\hat{c}_s\) and \(2s\)-stage collocation RKN methods based on \(\hat{c}_1,\dots ,\hat{c}_s,1+\hat{c}_1,\dots ,1+\hat{c}_s\). At \(n\)th integration step, the stage values of the \(2s\)-stage collocation RKN methods evaluated at \(t_n+(1+\hat{c}_1)h,\dots ,t_n+(1+\hat{c}_s)h\) can be used as the stage values of the collocation Gauss–Legendre RKN method for \((n+2)\)th integration step. In this way we obtain the corrector methods in which the integration processes can be proceeded two-step by two-step. The resulting parallel PC iteration methods which are called two-step-by-two-step (TBT) parallel-iterated RKN-type (PIRKN-type) PC methods based on Gauss–Legendre collocation points (two-step-by-two-step PIRKNG methods or TBTPIRKNG methods) give us a faster integration process. Fixed stepsize applications of these TBTPIRKNG methods to three widely used test problems show that the new parallel PC iteration methods are much more efficient when compared with the well-known parallel-iterated RKN methods (PIRKN methods) and sequential codes ODEX2 and DOP853 available from the literature.

A critical comparative assessment of differential equation-driven methods for structural topology optimization

In recent years, differential equation-driven methods have emerged as an alternate approach for structural topology optimization. In such methods, the design is evolved using special differential equations. Implicit level-set methods are one such set of approaches in which the design domain is represented in terms of implicit functions and generally (but not necessarily) use the Hamilton-Jacobi equation as the evolution equation. Another set of approaches are referred to as phase-field methods; which generally use a reaction-diffusion equation, such as the Allen-Cahn equation, for topology evolution. In this work, we exhaustively analyze four level-set methods and one phase-field method, which are representative of the literature. In order to evaluate performance, all the methods are implemented in MATLAB and studied using two-dimensional compliance minimization problems.

Further results of the estimate of growth of entire solutions of some classes of algebraic differential equations

Abstract In this article, by means of the normal family theory we estimate the growth order of entire solutions of some algebraic differential equations and improve the related results of Bergweiler, Barsegian, and others. We also estimate the growth order of entire solutions of a type system of a special algebraic differential equations. We give some examples to show that our results are sharp in special cases. Mathematica Subject Classification (2000): Primary 34A20; Secondary 30D35.

A linear homogeneous partial differential equation with entire solutions represented by Bessel polynomials

We have continued our earlier studies on entire solutions of some special type linear homogeneous partial differential equations. Specifically, we deal with entire solutions of the equations that are represented in convergent series of Bessel polynomials, and determine orders and types of the solutions, in terms of their Taylor coefficients, by establishing an analogue of Lindelöf–Pringsheim theorem as well as Wiman–Valiron type theory for such functions. Finally, by using value distribution theory of holomorphic functions, we are able to exhibit some uniqueness theorems of the entire (or meromorphic) solutions.

High‐Order Compact Implicit Difference Methods For Parabolic Equations in Geodynamo Simulation

A series of compact implicit schemes of fourth and sixth orders are developed for solving differential equations involved in geodynamics simulations. Three illustrative examples are described to demonstrate that high‐order convergence rates are achieved while good efficiency in terms of fewer grid points is maintained. This study shows that high‐order compact implicit difference methods provide high flexibility and good convergence in solving some special differential equations on nonuniform grids.

Explicit multi-step peer methods for special second-order differential equations

The construction of s-stage explicit two- and three-step peer methods of order p = 2 s and p = 3 s is considered for the solution of non-stiff second-order initial value problems where the right-hand side does not depend on y′. By numerical search with respect to large stability intervals and small error constants sequential and parallel methods are constructed. Numerical tests of these peer methods in MATLAB and comparisons with a Runge–Kutta–Nyström method show the efficiency of the proposed methods.

A NEW CLASS OF CHAOTIC SIMPLE THREE-NEURON CELLULAR NEURAL NETWORKS

In this letter we report a new class of chaotic three-neuron cellular neural networks that are described by special three-dimensional autonomous ordinary differential equations.

Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems

A family of polynomial differential systems describing the behavior of a chemical reaction network with generalized mass action kinetics is investigated. The coefficients and monomials are given by graphs. The aim of this investigation is to clarify the algebraic-discrete aspects of a Hopf bifurcation in these special differential equations. We apply concepts from toric geometry and convex geometry. As usual in stoichiometric network analysis we consider the solution set as a convex polyhedral cone and we intersect it with the deformed toric variety of the monomials. Using Gröbner bases the polynomial entries of the Jacobian are expressed in different coordinate systems. Then the Hurwitz criterion is applied in order to determine parameter regions where a Hopf bifurcation occurs. Examples from chemistry illustrate the theoretical results.

Duality for an indefinite inverse Sturm–Liouville problem

In this paper we solve an inverse spectral problem associated with a traditional Sturm–Liouville equation with an indefinite weight function. Specifically, we reconstruct a unique positive potential from two spectral sequences, given a weight function with a simple turning point in the interior of a finite interval with fixed end boundary conditions. We show the existence of special non-linear second order differential equations satisfied by these eigenvalue functions, dubbed the dual equations, eigenvalues whose asymptotics are used in the description of the reconstruction of the unknown potential.

A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations

Abstract A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer–Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations.

An improved class of generalized Runge–Kutta–Nyström methods for special second-order differential equations

We propose a new family of explicit methods of order four with two evaluations per step, for the numerical integration of special second-order differential equations given by y ″= f( y). These two-stage formulas can be seen as a generalization of the explicit two-stage Runge–Kutta–Nyström methods, providing better order and stability results. We will show that it is possible to obtain methods that are more efficient than the classical Runge–Kutta–Nyström one-step methods with the same number of evaluations per step, specially when highly oscillatory problems are considered. Some numerical experiments are discussed in order to show the good performance of the new schemes.

Explicit pseudo two-step RKN methods with stepsize control

This paper is devoted to variable stepsize strategy implementations of a class of explicit pseudo two-step Runge–Kutta–Nyström methods of arbitrarily high order for solving nonstiff problems for systems of special second-order differential equations. The constant stepsize explicit pseudo two-step Runge–Kutta–Nyström methods are developed into variable stepsize ones and equipped with embedded formulas giving a cheap error estimate for stepsize control. By two examples of widely-used test problems, a pseudo two-step Runge–Kutta–Nyström method of order 8 implemented with variable stepsize strategy is shown to be much more efficient than parallel and sequential codes available in the literature. With stringent error tolerances, this new explicit pseudo two-step Runge–Kutta–Nyström method is even superior to sequential codes in a sequential computer.

A Parallel ODE Solver Adapted to Oscillatory Problems

We propose a multistep method for solving special second-order ordinary differential equations with damped oscillatory solutions. The proposed methods integrate exactly (with only round-off error) ordinary polynomials and the product of trigonometric functions at a frequency ω by exponentials of a parameter g. When ωege0 they reduce to the classical Nystron and Cowell methods. Although there exist several methods with these properties, the proposed method allows independent computation of predictor and corrector which motivates parallel implementation.

First and second order linear dynamic equations on time scales

We consider first and second order linear dynamic equations on a time scale. Such equations contain as special cases differential equations, difference equationsq— difference equations, and others. Important properties of the exponential function for a time scale are presented, and we use them to derive solutions of first and second order linear dyamic equations with constant coefficients. Wronskians are used to study equations with non—constant coefficients. We consider the reduction of order method as well as the method of variation of constants for the nonhomogeneous case. Finally, we use the exponential function to present solutions of the Euler—Cauchy dynamic equation on a time scale.