Comparative Studies between Picard’s and Taylor’s Methods of Numerical Solutions of First Ordinary Order Differential Equations Arising from Real-Life Problems

To solve the first-order differential equation derived from the problem of a free-falling object and the problem arising from Newton’s law of cooling, the study compares the numerical solutions obtained from Picard’s and Taylor’s series methods. We have carried out a descriptive analysis using the MATLAB software. Picard’s and Taylor’s techniques for deriving numerical solutions are both strong mathematical instruments that behave similarly. All first-order differential equations in standard form that have a constant function on the right-hand side share this similarity. As a result, we can conclude that Taylor’s approach is simpler to use, more effective, and more accurate. We will contrast Rung Kutta and Taylor’s methods in more detail in the following section.

ASYMPTOTIC SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH TURNING POINTS

Boundary value problem; for third-order ordinary differential equations with turning points are studied as follows : epsilon gamma ' ' + f(x ; epsilon) gamma ' + g(x ; epsilon) gamma ' +h(x ; epsilon) gamma = 0 (- a < x < b, 0 epsilon 1), where f(x ; 0) has several multiple zero points in ( - n, b). the necessary conditions for exhibiting resonance is given, and the uniformly valid asymptotic solutions and the estimations of remainder terms are obtained.

Existence of Periodic Solutions for Odd Order Ordinary Differential Equations via the Homotopy Method

This paper deals with the problems of finding periodic solutions for the third order ordinary differential equations of the form (1) where T is a fixed positive number and f satisfies some additional conditions which will be stated later.The periodicity problem has been one of main topics in the qualitative theory of ordinary

Subspaces for weak mild solutions of the second order abstract differential equation

The topic on the subspaces for the polynomially or exponentially bounded weak mild solutions of the following abstract Cauchy problem d^2/（dr^2）u（t,x）=Au（t,x）;u（0,x）=x,d/（dt）u（0,x）=0,x∈X is studied, where A is a closed operator on Banach space X. The case that the problem is ill-posed is treated, and two subspaces Y（A, k） and H（A, ω） are introduced. Y（A, k） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（ t, x） such that ess sup{（1＋t）^-k｜d/（dt）〈v（t,x）,x^＊〉｜：t≥0,x^＊∈X^＊,｜x^＊‖≤1}〈＋∞. H（A, ω） is the set of all x in X for which the second order abstract differential equation has a weak mild solution v（t,x）such that ess sup{e^-ωl｜d/（dt）〈v（t,x）,x^＊）｜：t≥0,x^＊∈X^＊,‖x^＊‖≤1}〈＋∞. The following conclusions are proved that Y（A, k） and H（A, ω） are Banach spaces, and both are continuously embedded in X; the restriction operator A ｜ Y（A,k） generates a once-integrated cosine operator family { C（t） }t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖Y（A,k）≤M（1＋t）^k,arbitary t≥0; the restriction operator A ｜H（A,ω） generates a once- integrated cosine operator family {C（t）}t≥0 such that limh→0＋^-1/h‖C（t＋h）-C（t）‖H（A,ω）≤≤Me^ωt,arbitary t≥0.

HIGH ACCURACY FINITE VOLUME ELEMENT METHOD FOR TWO-POINT BOUNDARY VALUE PROBLEM OF SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, a high accuracy finite volume element method is presented for two-point boundary value problem of second order ordinary differential equation, which differs from the high order generalized difference methods. It is proved that the method has optimal order error estimate O(h3) in H1 norm. Finally, two examples show that the method is effective.

Existence of solutions of nonlinear two-point boundary value problems for 4nth-order nonlinear differential equation

Studies the existence of solutions of nonlinear two point boundary value problems for nonlinear 4n-th-order differential equationy (4n)=f(t,y,y′,y″,...,y (4n-1))(a)with the boundary conditions g 2i(y (2i)(a),y (2i+1)(a))=0,h 2i(y (2i)(c),y (2i+1)(c))=0,(i=0,1,...,2n-1)(b) where the functions f, g i and h i are continuous with certain monotone properties. For the boundary value problems of nonlinear nth order differential equationy (n)=f(t,y,y′,y″,...,y (n-1))many results have been given at the present time. But the existence of solutions of boundary value problem (a),(b) studied in this paper has not been covered by the above researches. Moreover, the corollary of the important theorem in this paper, i.e. existence of solutions of the boundary value problem.y (4n)=f(t,y,y′,y″,...,y (4n-1)) a 2iy (2i)(a)+a 2i+1y (2i+1)(a)=b 2i,c 2iy (2i)(c)+c 2i+1y (2i+1)(c)=d 2i,(i=0,1,...2n-1)has not been dealt with in previous works.

A PRIORI BOUNDS FOR PERIODIC SOLUTIONS OF A KIND OF THIRD-ORDER DELAY DIFFERENTIAL EQUATION

By means of continuation theorem of the coincidence degree theory, sufficient conditions are obtained for the existence of periodic solutions of a kind of third-order neutral delay functional differential equation with deviating arguments.

ON GLOBAL MEROMORPHIC SOLUTIONS OF SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH MEROMORPHIC COEFFICIENTS

The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.

Existence of positive solutions for fourth order singular differential equations with Sturm-Liouville boundary conditions

By using the upper and lower solutions method and fixed point theory,we investigate a class of fourth-order singular differential equations with the Sturm-Liouville Boundary conditions.Some sufficient conditions are obtained for the existence of C2[0,1] positive solutions and C3[0,1] positive solutions.

ON THE ASYMPTOTIC SOLUTIONS FOR A CLASS OF SECOND ORDER DIFFERENTIAL EQUATIONS WITH SLOWLY VARYING COEFFICIENTS

In this paper we study the asymptotic expansions of the solutions for a class of second order ordinary differential equations with slowly varying coefficients. The defect of the known works on these problems is noted, and the results in [1 - 4] are improved and extended by means of the modified method of multiple scales.

The Growth Orders of Meromorphic Solutions of Differential Equations

By meas of the Nevanlinna theory of the value distribution of meromorphic functions, this paper discusses the orders of growth of meromorphic solutions of differential equation and proves that the form of the solution is determined if the order are sufficiently large.

Existence and Multiplicities of Solutions for Asymptotically Linear Ordinary Differential Equations Satisfying Sturm-Liouville BVPs with Resonance

In this paper, we prove existence and multiplicities of solutions for asymptotically linear ordinary differential equations satisfying Sturm-Liouville boundary value conditions with resonance. Adding assumption H3 that is similar to (LL) in Theorem 1.1, by index theory and Morse theory, we obtain more nontrivial solutions.

Periodic Solutions of Even Order Nonlinear Differential Equations

This paper deals with even order nonlinear differential equations. Applying the Schauder fixed point theorem and the Schwartz inequality technique, we give a criterion to ensure the existence and uniqueness of periodic solutions. This criterion is a complement of the well known Lazer type results.

UNIQUENESS OF SOLUTIONS OF SINGULARLY PERTURBED BOUNDARY VALUE PROBLEMS FOR THIRD QRDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ε>0: where i=1, 2; a(?)(ε), β(ε) and γ(ε) are functions defined on (0, ε_o], while ε_o>0 is a constant.This paper is the continuation of our works [4, 6].

ON HYPER-ORDER OF MEROMORPHIC SOLUTIONS OF FIRST-ORDER ALGEBRAIC DIFFERENTIAL EQUATIONS

The authors give a precise estimate of the hyper-order of meromorphic solutions of general first-order algebraic differential equations.

Hojman's theorem of the third-order ordinary differential equation

This paper extends Hojman＇s conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results.

Criteria for System of Three Second-Order Ordinary Differential Equations to Be Reduced to a Linear System via Restricted Class of Point Transformation

This paper is devoted to the study of the linearization problem of system of three second-order ordinary differential equations and . The necessary conditions for linearization by general point transformation and are found. The sufficient conditions for linearization by restricted class of point transformation and are obtained. Moreover, the procedure for obtaining the linearizing transformation is provided in explicit forms. Examples demonstrating the procedure of using the linearization theorems are presented.

ON THE BOUNDEDNESS AND PERIODICITY OF THESOLUTIONS OF A CERTAIN VECTOR DIFFERENTIALEQUATION OF THIRD-ORDER

There are given sufficient conditions for the ultimate boundedness of solutions and for the existence of periodic solutions of a certain vector differential equation of third-order.

Solution of Second-Order Ordinary Differential Equations via Simulated Annealing

In this paper, we approach the problem of obtaining approximate solution of second-order initial value problems by converting it to an optimization problem. It is assumed that the solution can be approximated by a polynomial. The coefficients of the polynomial are then optimized using simulated annealing technique. Numerical examples with good results show the accuracy of the proposed approach compared with some existing methods.

Result on the Convergence Behavior of Solutions of Certain System of Third-Order Nonlinear Differential Equations

Convergence behaviors of solutions arising from certain system of third-order nonlinear differential equations are studied. Such convergence of solutions corresponding to extreme stability of solutions when relates a pair of solutions of the system considered. Using suitable Lyapunov functionals, we prove that the solutions of the nonlinear differential equation are convergent. Result obtained generalizes and improves some known results in the literature. Example is included to illustrate the result.