# A One-Point Third-Derivative Hybrid Multistep Technique for Solving Second-Order Oscillatory and Periodic Problems

Authors: Mufutau Ajani Rufai; Ali Shokri; Ezekiel Olaoluwa Omole
Journal: Journal of Mathematics
 (2023)

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## Abstract

This paper describes a third-derivative hybrid multistep technique (TDHMT) for solving second-order initial-value problems (IVPs) with oscillatory and periodic problems in ordinary differential equations (ODEs), the coefficients of which are independent of the frequencyomega and step size h. This research is significant because it has numerous applications to real-life phenomena such as chaotic dynamical systems, almost periodic problems, and duffing equations. The current method is derived from the collocation of a derivative function at the equidistant grid and off-grid points. The TDHMT obtained is a continuous scheme for obtaining simultaneous approximations to the solution and its derivative at each point in the x0,xN interval integration. The presence of high derivatives increases the order of the method, which increases the accuracy method’s order and the stability property, as discussed in detail. Finally, the proposed method is compared to existing methods in the literature on some oscillatory and periodic test problems to demonstrate the technique’s effectiveness and productivity.


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## Body

## 1. Introduction

The numerical solution of general second-order IVPs of ODEs of the form (1) is the focus of this study.(1)u′′x=fx,u,u′,ux0=u0,u′x0=u0′,u,x,f∈ℝd,whose solution u(x) is assumed to have an oscillatory or periodic behaviour.Problems with structure (1) arise much of the time in astrophysics and space, nuclear physics, celestial mechanics, molecular dynamics, circuit theory, chemical kinetics, and other various areas of applications in science and engineering. The study places emphasis on second-order IVPs, where the first-derivative does not show up explicitly, due to their applications, and many of these problems do not have known analytical solutions. Because of real-life applications of this problem, numerous scientists are propelled to examine its numerical solutions (see [1, 2] and references therein).The authors of reference [1] developed a 3-point variable step block hybrid method (3-point VSBHM) using Lagrange polynomials as the basis function. The 3-point VSBHM was applied to difficult chemical problems such as the Belousov–Zhabotinsky reaction and Hires. It has been demonstrated that the method meets the basic requirement for convergence. Achar tenders symmetric multistep Obrechkoff methods with eight-order algebraic accuracy and zero-phase lag for periodic initial-value problems of second-order differential equations in reference [2]. The method was used to solve practical problems of the form (1), and the solution demonstrates how the problems behave when used to solve periodic initial-value problems.In addition, Wu et al. in reference [3] presented a novel method for simulating turbulent reactive flows with stiff chemistry called efficient time-stepping techniques. The method was used to solve various types of stiff problems in chemistry, specifically turbulent reactive flows, and its applicability was demonstrated. Skwame et al. developed an equidistant one-step block hybrid method for treating second-order ordinary differential equations. The developed one-step block hybrid method was applied to a real-world physics and engineering problem, namely, the cooling of a body in reference [4]. Next, the authors of reference [5] proposed a new trigonometrically two-step hybrid method for solving second-order ODEs with one or two frequencies. The methods were examined and discussed in terms of their practical application. Meanwhile, Liu et al. [6] considered a new modified embedded 5(4) pair of explicit Runge–Kutta methods for numerically solving the Schrödinger equation. The methods were investigated and applied to the Schrodinger equation, with the solution illustrated in curves.In references [7, 8], the effects of Soret and Dufour on heat and mass transfer of boundary layer flow over porous wedge with thermal radiation were investigated. The investigated governing equations have boundary conditions that are nondimensionalized by introducing some nondimensional variables. The following equations are solved numerically using the bivariate spectral relaxation method (BSRM). This type of ODE has a wide range of applications in engineering, including oil bed recovery, filtration, thermal insulations, heat exchangers, and geothermal analysis. Kamran et al. [9] propounded a numerical simulation of the time fractional BBM-Burger equation using cubic B-spline functions. The authors of reference [10] investigated an asymptotic reduction of the dimension of the solution space for dynamical systems. The authors of reference [11] studied the Cauchy problem for the degenerate parabolic convolution equation. In the work of reference [12], the mathematical and numerical modeling of coupled dynamic thermoelastic problems for isotropic bodies was investigated. The authors of reference [13] presented an inverse boundary value problem for the Boussinesq-love equation with a nonlocal integral condition in his paper. Later, Prasad and Isik [14] discuss numerical solutions to boundary value problems using contractions.Furthermore, some reports on recent articles for problem (1) resolution will be explored. According to Franco and Gomez, reference [15] has developed trigonometrically fitted nonlinear two-step methods. In reference [16], Kalogiratou also presented diagonally implicit trigonometrically fitted symplectic Runge–Kutta methods. In references [17] and [18], Tsitouras and Simos have derived and implemented high phase-lag order four-step and explicit Numerov-type methods. More recently, Shokri has constructed a new eight-order symmetric two-step multiderivative technique in reference [19].It should be noted that the majority of the methods described mentioned above do not use the first-derivative values in their derivations. Consequently, this study proposes and investigates a new third-derivative hybrid multistep method for numerically solving the problem in equation (1). The new TDHMT introduced and used first-derivative values in its formulation, giving it a broader application space and higher computational productivity than existing multistep type strategies. Lastly, the TDHMT is written in a block-by-block fashion, and approximations are provided at every grid point of the integration interval at the same time, resulting in a lower total number of function evaluations.


## 2. Mathematical Formulation of the TDHMT

This section’s goal is to derive a TDHMT with two off-grid points inxn,xn+2. To do so, we assume that the exact solution ux is approximated by the polynomial μx,(2)ux≃μx=∑n=07bnxn,where bn∈ℝ are real unknown coefficients that are resolved after taking collocation and interpolation conditions into account at selected points. Let xn+r=xn+1/2h,xn+s=xn+3/2h be the two off-grid points on xn,xn+2 and the approximation in equation (2), and its first derivatives applied to the point xn, its second derivative applied to the points xn,xn+r,xn+1,xn+s,xn+2, and its third derivative applied to the points xn+2. Then, we get a system of 8 equations with 8 unknowns bn,n=017, given by(3)μxn=un,μ′xn=un′,μ′′xn=fn,μ′′′xn+2=gn+2,μ′′xn+r=fn+r,μ′′xn+1=fn+1,μ′′xn+s=fn+s,μ′′xn+2=fn+2.After obtaining the values of the coefficientsbn,n=017, substituting its values into equation (2), and changing the variable, x=xn+zh, then, the polynomial in equation (2) maybe written as follows:(4)uxn+zh=α0zun+hα1zun′+h2β0zfn+βrzfn+r+β1zfn+1+βszfn+s+β2zfn+2+h3γ1zgn+2,where h is the chosen step-size and α0z,α1z,β0z,βrz,β1z,βsz,β2z,γ1z are continuous coefficients.


### 2.1. Main Formulas of the TDHMT

The main formulas are generated by substituting the values ofαjz, βjz, j=0,1,2, βrz, βsz, and γ1z into equation (4) and evaluating μx and μ′x at the point xn+2=xn+2h to obtain the following approximations for uxn+2 and u′xn+2.(5)un+2=h231fn105+8fn+1105+1088945fn+r+64105fn+s−25fn+2189+263h3gn+2+2hun′+un,un+2′=h7fn45+4fn+115+7fn+245+3245fn+r+3245fn+s+un′.


### 2.2. Additional Formulas of the TDHMT

The additional formulas are obtained by evaluating ofμx and μ′x at the points xn+r, xn+1, and xn+s. Thus, the following additional formulas that form the proposed method, namely, TDHMT was obtained.(6)un+r=h23247fn53760+337fn+r3024+41560fn+s−1193fn+113440−15139fn+2483840+107h3gn+216128+hun′2+un,un+1=h239fn280+418945fn+r+635fn+s−5fn+128−563fn+27560+163h3gn+2+hun′+un,un+s=h23891fn17920+447560fn+r+33112fn+s−297fn+14480−2127fn+217920+45h3gn+21792+3hun′2+un,un+r′=h1873fn11520+2345fn+r+47180fn+s−311fn+1960−1277fn+211520+3128h2gn+2+un′,un+1′=h37fn240+fn+120+98135fn+r+215fn+s−137fn+22160+172h2gn+2+un′,un+s′=h201fn1280+99fn+1320+710fn+r+920fn+s−149fn+21280+3128h2gn+2+un′.


## 2.1. Main Formulas of the TDHMT

The main formulas are generated by substituting the values ofαjz, βjz, j=0,1,2, βrz, βsz, and γ1z into equation (4) and evaluating μx and μ′x at the point xn+2=xn+2h to obtain the following approximations for uxn+2 and u′xn+2.(5)un+2=h231fn105+8fn+1105+1088945fn+r+64105fn+s−25fn+2189+263h3gn+2+2hun′+un,un+2′=h7fn45+4fn+115+7fn+245+3245fn+r+3245fn+s+un′.


## 2.2. Additional Formulas of the TDHMT

The additional formulas are obtained by evaluating ofμx and μ′x at the points xn+r, xn+1, and xn+s. Thus, the following additional formulas that form the proposed method, namely, TDHMT was obtained.(6)un+r=h23247fn53760+337fn+r3024+41560fn+s−1193fn+113440−15139fn+2483840+107h3gn+216128+hun′2+un,un+1=h239fn280+418945fn+r+635fn+s−5fn+128−563fn+27560+163h3gn+2+hun′+un,un+s=h23891fn17920+447560fn+r+33112fn+s−297fn+14480−2127fn+217920+45h3gn+21792+3hun′2+un,un+r′=h1873fn11520+2345fn+r+47180fn+s−311fn+1960−1277fn+211520+3128h2gn+2+un′,un+1′=h37fn240+fn+120+98135fn+r+215fn+s−137fn+22160+172h2gn+2+un′,un+s′=h201fn1280+99fn+1320+710fn+r+920fn+s−149fn+21280+3128h2gn+2+un′.


## 3. Analysis of the TDHMT

This section analyzes the characteristics of the proposed TDHMT, which is based on an extension of Dahlquist barrier for multistep methods.


### 3.1. Accuracy and Consistency of the TDHMT

The TDHMT method given in equations (5) and (6) might be formulated as follows:(7)PVn=hQVn′+h2RFn+h3SGn,where P,Q,R,S denote matrices of coefficients in equations (5) and (6), and(8)Vn=un,un+r,un+1,un+s,un+2T,Vn′=un′,un+r′,un+1′,un+s′,un+2′T,Fn=fn,fn+r,fn+1,fn+s,fn+2T,Gn=gn,gn+r,gn+1,gn+s,gn+2T.Using [20], we assume that ux is a sufficiently differentiable function, we define the operator ℓ as follows:(9)ℓux;h=∑j∈Jα¯juxn+jh−hβ¯ju′xn+jh−h2γ¯ju′′xn+jh−h3μ¯ju′′′xn+jh,where α¯j,β¯j, γ¯j,μ¯j are the vector columns of the matrices P,Q,R,S, respectively, and J=0,r,1,s,2. By using the Taylor series about xn, we expand the formulas in equation (9) and obtain as follows:(10)ℓux;h=C¯0uxn+C¯1hu′xn+C¯2h2u′′xn+…+C¯qhquqxn+…,with(11)C¯q=1q!∑j∈Jkjqα¯j−q∑j∈Jkjq−1β¯j−qq−1∑j∈Jkjq−2γ¯j,where q=0,1,2,…. The abovementioned operator and the associated formulas are said to be of order p, according to reference [21], if C¯0=C¯1=…=C¯p+1=0,C¯p+2≠0. C¯i is the column vectors and C¯p+2 is the vector of local truncation errors. For the proposed TDHMT in equations (5) and (6), we have C¯0=0,C¯1=0,C¯2=0,…,C¯7=0 and(12)C¯8=−17560,−115120,−78130965760,−116128,−1111146880,3373870720,−116128,−11143360T.Therefore, the proposed TDHMT has six algebraic order of convergences. Since the order of the new method is not≤1, then the TDHMT is also consistent.


### 3.2. Zero-Stability and Convergence of the TDHMT

Zero-stability of the TDHMT is concerned with the behavior of the difference scheme (7) in the limit as h tends to zero. For h⟶0, (7) may be written as follows:(13)P0V¯u−P1V¯u−1=0,where(14)V¯u=un+2,un+s,un+1,un+rT,V¯u−1=un,un+s−2,un−1,un+r−2T,P0=1000010000100001,P1=1000100010001000.The proposed TDHMT is said to be zero-stable if the rootspj of the first characteristic polynomial ρp denoted by ρp=detP0p−P1, satisfy pj≤1, and those roots with pj=1 have a multiplicity which is not greater than 2 (see [22] for more details). Since ρp=p−1p3, the TDHMT is zero stable. The requirements for a multistep method to be convergent are that it must be zero-stable and consistent. Because these two conditions are met, the TDHMT is also convergent.


### 3.3. Linear Stability Analysis of the TDHMT

The following test equation to determine the linear stability of the newly constructed strategy is considered.(15)u′′x=−2νu′x−ν2ux,λ>0.The TDHMT is applied to the scalar test (15), yielding the matrix difference equation shown.(16)Aun+run+1un+sun+2un+r′un+1′un+s′un+2′=Bun+r−2un−1un+s−2unun+r−2′un−1′un+s−2′μn′,with(17)A=337ν23024+1−1193ν21344041ν2560−15139ν2483840000107ν216128418ν29451−5ν2286ν235−563ν27560000ν263447ν2560−297ν2448033ν2112+1−2127ν21792000045ν217921088ν29458ν210564ν21051−25ν21890002ν26323ν245−1960311ν247ν2180−1277ν2115201003ν212898ν2135ν2202ν215−137ν22160010ν2727ν21099ν23209ν220−149ν212800013ν212832ν2454ν21532ν2457ν2450001,B=0001−3247ν253760000120001−39ν228000010001−3891ν217920000320001−31ν21050002000−1873ν2115200001000−124037ν20001000−201ν212800001000−1457ν20001.The eigenvalues of the stability matrixMz=A−1B,z=νh are used to investigate the boundedness of their solutions. If ν∈ℝ, the entire absolute stability region then gets smaller to a real interval known as an interval of periodicity. This interval is obtained as 0,∞. Figure 1 displays the TDHMT method stability region.Figure 1
Stability region of the proposed TDHMT.


## 3.1. Accuracy and Consistency of the TDHMT

The TDHMT method given in equations (5) and (6) might be formulated as follows:(7)PVn=hQVn′+h2RFn+h3SGn,where P,Q,R,S denote matrices of coefficients in equations (5) and (6), and(8)Vn=un,un+r,un+1,un+s,un+2T,Vn′=un′,un+r′,un+1′,un+s′,un+2′T,Fn=fn,fn+r,fn+1,fn+s,fn+2T,Gn=gn,gn+r,gn+1,gn+s,gn+2T.Using [20], we assume that ux is a sufficiently differentiable function, we define the operator ℓ as follows:(9)ℓux;h=∑j∈Jα¯juxn+jh−hβ¯ju′xn+jh−h2γ¯ju′′xn+jh−h3μ¯ju′′′xn+jh,where α¯j,β¯j, γ¯j,μ¯j are the vector columns of the matrices P,Q,R,S, respectively, and J=0,r,1,s,2. By using the Taylor series about xn, we expand the formulas in equation (9) and obtain as follows:(10)ℓux;h=C¯0uxn+C¯1hu′xn+C¯2h2u′′xn+…+C¯qhquqxn+…,with(11)C¯q=1q!∑j∈Jkjqα¯j−q∑j∈Jkjq−1β¯j−qq−1∑j∈Jkjq−2γ¯j,where q=0,1,2,…. The abovementioned operator and the associated formulas are said to be of order p, according to reference [21], if C¯0=C¯1=…=C¯p+1=0,C¯p+2≠0. C¯i is the column vectors and C¯p+2 is the vector of local truncation errors. For the proposed TDHMT in equations (5) and (6), we have C¯0=0,C¯1=0,C¯2=0,…,C¯7=0 and(12)C¯8=−17560,−115120,−78130965760,−116128,−1111146880,3373870720,−116128,−11143360T.Therefore, the proposed TDHMT has six algebraic order of convergences. Since the order of the new method is not≤1, then the TDHMT is also consistent.


## 3.2. Zero-Stability and Convergence of the TDHMT

Zero-stability of the TDHMT is concerned with the behavior of the difference scheme (7) in the limit as h tends to zero. For h⟶0, (7) may be written as follows:(13)P0V¯u−P1V¯u−1=0,where(14)V¯u=un+2,un+s,un+1,un+rT,V¯u−1=un,un+s−2,un−1,un+r−2T,P0=1000010000100001,P1=1000100010001000.The proposed TDHMT is said to be zero-stable if the rootspj of the first characteristic polynomial ρp denoted by ρp=detP0p−P1, satisfy pj≤1, and those roots with pj=1 have a multiplicity which is not greater than 2 (see [22] for more details). Since ρp=p−1p3, the TDHMT is zero stable. The requirements for a multistep method to be convergent are that it must be zero-stable and consistent. Because these two conditions are met, the TDHMT is also convergent.


## 3.3. Linear Stability Analysis of the TDHMT

The following test equation to determine the linear stability of the newly constructed strategy is considered.(15)u′′x=−2νu′x−ν2ux,λ>0.The TDHMT is applied to the scalar test (15), yielding the matrix difference equation shown.(16)Aun+run+1un+sun+2un+r′un+1′un+s′un+2′=Bun+r−2un−1un+s−2unun+r−2′un−1′un+s−2′μn′,with(17)A=337ν23024+1−1193ν21344041ν2560−15139ν2483840000107ν216128418ν29451−5ν2286ν235−563ν27560000ν263447ν2560−297ν2448033ν2112+1−2127ν21792000045ν217921088ν29458ν210564ν21051−25ν21890002ν26323ν245−1960311ν247ν2180−1277ν2115201003ν212898ν2135ν2202ν215−137ν22160010ν2727ν21099ν23209ν220−149ν212800013ν212832ν2454ν21532ν2457ν2450001,B=0001−3247ν253760000120001−39ν228000010001−3891ν217920000320001−31ν21050002000−1873ν2115200001000−124037ν20001000−201ν212800001000−1457ν20001.The eigenvalues of the stability matrixMz=A−1B,z=νh are used to investigate the boundedness of their solutions. If ν∈ℝ, the entire absolute stability region then gets smaller to a real interval known as an interval of periodicity. This interval is obtained as 0,∞. Figure 1 displays the TDHMT method stability region.Figure 1
Stability region of the proposed TDHMT.


## 4. Implementation Details

In this part, the discussion of the step-by-step procedure for implementation of the new method tagged TDHMT is fully stated. The TDHMT is implemented in a fashion-by-fashion block method to simultaneously generate the solution at the initial point to the terminal point. The method is self-starting and does not require any separate predictors. In fact, each block integrators in equations (5) and (6) form a system of equations which are applied along with Newton’s method. The derived starting values, also known as predictors, used Newton’s method and are considered approximations provided by the Taylor series expansion formulas.(18)un+i=un+hun′+h22fn+h36gn,un+i′=un′+hfn+h22gn,i=012…,k.gn+i,i=01/2…,k, occurring in equations (5) and (6), denotes the third derivative at xn+i,i=01/2…,k. In other to obtain a closed form solution of equations (5) and (6) which is expanded in equation (18), it is important to calculate the values of gn+i,i=01/2…,k. For more clarification,(19)u′′′x=dwx,u,u′dx,i=012…,k,u′′′x=dwdx+dwdyy′+dwdu′w,i=012…,k.


## 5. Numerical Examples and Discussion of the Results

This section is concerned with the presentation of the computational results as well as the CPU time required to compute the results, and the number of function evaluation. The following notations will be used for simplicity presentation.(i)
NFE—Number of function evaluation(ii)
TDHMT—The one-pointthird-derivative hybrid multistep technique of order 6: newly proposed method(iii)
SAMTD—The multiderivative method of order 12 produced by reference [19](iv)
SIMOSMTD—The 12th order multiderivative Obrechkoff Method developed by reference [23](v)
VDMTD—The 12th order Obrechkoff ultiderivative method contructed by reference [24](vi)
AMTD—The 8th order Obrechkoff multiderivative method introduced by reference [2](vii)
WETMTD—The Obrechkoff multiderivative method of order 12 presented by reference [25](viii)
SKMTD—The four-step multiderivative method of order 8 studied by reference [26]Example 1.
The following is the nonlinear undamped Duffing’s equation(20)y′′=−y−y3+Bcosωx,y0=0.200426728067,y′0=0,where B=0.002, ω=1.01, and x∈0,40.5π/1.01. The exact solution for equation (20) gx=∑i=03K2i+1cos2i+1ωx, where(21)K1,K3,K5,K7=0.200179477536,0.246946143×10−3,0.304016×10−6,0.374×10−9.
The absolute errors atx=40.5π/1.01 are given in Table 1, where M=40.5π/1.01 and the CPU times are listed in Table 2, while Table 3 shows the comparison of the number of functions evaluation (NFE).Table 1
Comparison of the absolute errors at the end point of TDHMT with absolute error of other existing methods, namely, SKMTD, SAMTD, SIMOSMTD, VDMTD, AMTD, and WETMTD for Example1.
hTDHMTSKMTDSAMTDSIMOSMTDVDMTDAMTDWETMTDM/5003.14e−108.26e−119.31e−113.15e−044.06e−054.09e−054.08e−05M/10005.23e−129.52e−128.03e−121.81e−051.87e−061.27e−061.27e−06M/20008.43e−143.74e−125.52e−121.07e−063.84e−083.94e−083.93e−08M/30005.02e−157.16e−137.25e−122.09e−075.13e−095.18e−095.17e−09M/40001.31e−152.18e−136.99e−126.55e−083.19e−091.23e−091.23e−09M/50003.48e−169.65e−146.65e−122.67e−089.89e−104.09e−104.07e−10Table 2
Comparison of the CPU time of TDHMT with the CPU time of other existing methods, namely, SKMTD, SAMTD, SIMOSMTD, VDMTD, AMTD, and WETMTD for Example1.
hTDHMTSKMTDSAMTDSIMOSMTDVDMTDAMTDWETMTDM/5002.151.421.451.441.481.191.41M/10003.542.252.872.892.942.312.89M/20005.245.126.276.236.364.816.24M/30008.147.619.869.869.727.559.55M/400011.3810.2513.4213.5513.399.9913.06M/500015.0214.8116.8616.9216.9712.8616.50Table 3
Comparison of the NFE of TDHMT with other existing methods, namely, SKMTD, SAMTD, SIMOSMTD, VDMTD, AMTD, and WETMTD for Example1.
hTDHMTSKMTDSAMTDSIMOSMTDVDMTDAMTDWETMTDM/5001468149015011550150016001620Example 2.
The initial-value problem(22)y′′=−100y+99sinx,y0=1,y′0=11.
The exact solutionyt=sint+sin10t+cos10t. This equation has been solved numerically for 0≤x≤10π using exact starting values. The step lengths h=π/50, h=π/100, h=π/200, h=π/300, h=π/400, and h=π/500.Example 3.
Consider the initial-value problem(23)y′′=8y21+2x,y0=1,y′0=−2,x∈0,4.5,with the exact solution(24)yx=11+2x.
The absolute errors atx=4.5 are given in Table 4, the CPU times are listed in Table 5, and the NFE is shown in Table 6.Table 4
Comparison of the absolute errors at the end point of TDHMT with absolute error of other existing methods, namely, SIMOSMTD and AMTD for Example3.
hTDHMTSIMOSMTDAMTD4.5/5002.73e−191.07e−097.46e−144.5/10004.71e−219.24e−125.36e−164.5/20007.46e−235.45e−142.88e−184.5/30006.16e−242.45e−151.25e−194.5/40009.03e−252.63e−161.32e−204.5/50002.37e−264.60e−172.29e−21Table 5
Comparison of the CPU time of TDHMT with the CPU time of other existing methods, namely, SIMOSMTD and WETMTD for Example3.
hTDHMTSIMOSMTDAMTD4.5/5000.340.360.194.5/10000.540.620.764.5/20001.081.231.204.5/40001.631.891.624.5/50002.272.592.06Table 6
Comparison of the NFE of TDHMT with the NFE of other existing methods, namely, SIMOSMTD and WETMTD for Example3.
hTDHMTSIMOSMTDAMTD4.5/500160017501900Example 4.
Franco and Palacios investigated an almost periodic orbital problem:(25)y″=ϵeiψx,y0=1,y′0=i,y∈ℂ,or equivalently by(26)u′′+u=ϵcosψx,u0=1,u′0=0,v′′+v=ϵsinψx,v0=0,v′0=1,ϵ=0.001, ψ=0.01. The theoretical solution is as follows:(27)yx=ux+ivx,u,v∈ℝ,where(28)ux=1−ϵ−ψ21−ψ2cosx+ϵ1−ψ2cosψx,vx=1−ϵψ−ψ2−ψ2sinx+ϵ1−ψ2sinψx.
Forxin0,1000pi, this system of equations has been solved. Table 7 shows the absolute errors at the end point, Table 8 shows the CPU times, and Table 9 presents the NFE where M=40.5π/1.01.Table 7
Comparison of the absolute errors at the end point of TDHMT with absolute error of other existing methods, namely, SIMOSMTD, VDMTD, and AMTD for Example4.
hTDHMTSIMOSMTDVDMTDAMTDM/5002.75e−82.04e−065.19e−061.07e−04M/10005.71e−102.40e−071.59e−071.52e−05M/15001.06e−115.87e−089.03e−081.12e−06M/20007.75e−122.64e−081.45e−084.96e−07M/25002.71e−131.60e−082.43e−091.25e−07M/30009.78e−141.13e−081.82e−093.45e−08Table 8
Comparison of the CPU time of TDHMT with the CPU time of other existing methods, namely, SIMOSMTD, VDMTD, and AMTD for Example4.
hTDHMTSIMOSMTDVDMTDAMTDM/5000.320.360.420.37M/10000.680.760.800.78M/15001.091.171.191.20M/20001.451.591.611.58M/25001.731.952.001.98M/30002.212.372.342.39Table 9
Comparison of the NFE of TDHMT with the NFE of other existing methods, namely, SIMOSMTD, VDMTD, and AMTD for Example4.
hTDHMTSIMOSMTDVDMTDAMTDM/500780820900890Example 5.
The almost periodic orbital problem studied by Stiefel and Bettis:(29)y′′+y=0.001eix,y0=1,y′0=0.9995i,y∈ℂ,or equivalently by(30)u′′+u=0.001cosψx,u0=1,u′0=0,v′′+v=0.001sinψx,v0=0,v′0=0.9995.
The theoretical solution:yx=ux+ivx, where u,v∈ℝ and(31)ux=cosx+0.0005sinx,vx=sinx−0.0005xcosx.
This system has been solved forx∈0,1000π.


## 6. Concluding Remarks

This study has successfully derived, analyzed, and implemented a third-derivative hybrid multistep technique (TDHMT) for the direct solution of second-order initial-value problems (IVPs) in ODEs whose coefficients do not depend on the frequency (omega) and step-size h in this study. The absolute error of the TDHMT, the CPU time compared with other existing methods, and the NFE compared with other existing methods in the literature, respectively, are shown in Tables 1–15. The TDHMT’s numerical results show that it is more efficient, accurate, capable, and stable in solving oscillatory and periodic problems with lesser NFEs.Table 10
Comparison of the absolute errors at the end point of TDHMT with absolute error of other existing methods, namely, SIMOSMTD, VDMTD, AMTD, and WETMTD for Example2.
hTDHMTSIMOSMTDVDMTDAMTDWETMTDπ/502.32e−253.03e−063.64e−062.23e−081.76e−16π/1004.71e−271.15e−086.79e−097.98e−114.54e−20π/2008.05e−294.50e−111.07e−125.24e−141.92e−24π/3004.23e−301.76e−128.13e−156.38e−164.65e−27π/4009.83e−311.76e−132.56e−162.74e−176.34e−29π/5002.12e−322.95e−141.79e−172.38e−182.25e−30Table 11
Comparison of the CPU time of TDHMT with the CPU time of other existing methods, namely, SIMOSMTD, VDMTD, AMTD, and WETMTD for Example2.
hTDHMTSIMOSMTDVDMTDAMTDWETMTDπ/500.160.170.250.190.11π/1000.480.510.530.450.28π/2001.770.860.830.750.58π/3001.081.141.150.950.92π/4001.251.391.401.231.26π/5001.631.701.781.471.56Table 12
Comparison of the NFE of TDHMT with the NFE of other existing methods, namely, SIMOSMTD, VDMTD, AMTD, and WETMTD for Example2 at the end point.
hTDHMTSIMOSMTDVDMTDAMTDWETMTDπ/50480790850800930Table 13
Comparison of the absolute errors at the end point of TDHMT with absolute error of other existing methods, namely, SIMOSMTD, VDMTD, and AMTD for Example5.
hTDHMTSIMOSMTDVDMTDAMTDM/5004.16e−051.86e−031.91e−032.55e−02M/10007.71e−074.32e−047.44e−044.94e−02M/15002.17e−081.08e−041.89e−058.74e−04M/20008.35e−092.69e−052.61e−071.21e−05M/25004.03e−106.70e−064.48e−091.96e−07M/30009.78e−111.46e−068.55e−114.36e−09Table 14
Comparison of the CPU time of TDHMT with the CPU time of other existing methods, namely, SIMOSMTD, VDMTD, and AMTD for Example5.
hTDHMTSIMOSMTDVDMTDAMTDM/5000.050.050.060.05M/10000.110.110.120.11M/15000.310.310.300.30M/20000.680.720.620.80M/25001.361.471.341.39M/30002.652.712.642.66Table 15
Comparison of the NFE of TDHMT with the NFE of other existing methods, namely, SIMOSMTD, VDMTD, and AMTD for Example5.
hTDHMTSIMOSMTDVDMTDAMTDM/500350400500450



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*Source: 2343215-2023-01-23.xml*