An Efficient Zero-Stable Numerical Method for Fourth-Order Differential Equations

Abstract

The purpose of this paper is to produce an efficient zero-stable numerical method with the same order of accuracy as that of the main starting values (predictors) for direct solution of fourth-order differential equations without reducing it to a system of first-order equations. The method of collocation of the differential system arising from the approximate solution to the problem is adopted using the power series as a basis function. The method is consistent, symmetric, and of optimal order 𝑝=6. The main predictor for the method is also consistent, symmetric, zero-stable, and of optimal order 𝑝=6.

1. Introduction

Many problems leading to fourth-order differential equations of the form 𝑦^𝑖𝑣=𝑓𝑡,𝑦,𝑦¹,𝑦^,𝑦^,𝑦^(𝑚−1)𝑡⁰=𝜇^𝑚−1,𝑚=1,2,3,4,(1.1) are of great interest to scientists and engineers formulating mathematical models for elastic bodies. Investigators, both theoretical and numerical analysts, have concerned themselves with the study and solutions of such equations. However, only a limited number of analytical methods are available for solving (1.1) directly without reduction to a first-order system of initial value problems (ivps) of type 𝑦^=𝑓(𝑡,𝑦),𝑦𝑡₀=𝜇,𝑓∈𝐶¹[𝑎,𝑏],𝑡∈𝑅,𝑦∈𝑅^𝑚.(1.2) The many numerical methods adopted for such higher-order differential equations (1.1) are only for handling first-order equations (1.2), [1–4].

The reduction of such problems of type (1.1) to systems of first-order equations, to our knowledge, leads to serious computational burden as well as wastage in computer time. These setbacks had been addressed by some authors [5–7]. In recent times, attempts have been made to formulate numerical algorithms capable of solving problem (1.1) directly using differing methods of derivation with varying degree of accuracy [8–12]. These attempts are not without their associated limitations, some of which include low stepnumber 𝑘, too many function evaluations, and the combination of lower-order predictors with the correctors in the predictor-corrector methods [13, 14]. All these have effects on the accuracy of the methods.

The proposed zero-stable order six methods are designed to have higher step number and reduced function evaluations, in order to address these observed problems. The derived method is capable of handling problems where 𝑓 is either linear or nonlinear. The numerical results of the method are compared with an existing method [3].

2. Description of the Method

Let the approximate solution to problem (1.1) be a partial sum of a power series of the form: 𝑦(𝑥)=^𝑘+2_𝑗=0𝑎_𝑗𝑥^𝑗.(2.1) Taking the fourth derivative of (2.1) and using this in (1.1) yields^𝑘+2_𝑗=4𝑗(𝑗−1)(𝑗−2)(𝑗−3)𝑎_𝑗𝑥^𝑗−4=𝑓𝑥,𝑦,𝑦^,𝑦^,𝑦^.(2.2) To be able to obtain an open method with three function evaluations, collocation points are taken in (2.2) at all odd grid points 𝑥=𝑥_𝑛+𝑖,𝑖=1(2)𝑘. Equation (1.2) is interpolated at all grid points 𝑥=𝑥_𝑛+𝑖,𝑖=0(1)𝑘−1,𝑥∈𝑅, where 𝑅 is the set of real numbers. The choice of number of interpolation points was guided by the intended order of the method as well as the number of collocation points.

The system of equations obtained from the collocation and interpolation above is represented by the matrix equation:𝐴𝑋=𝐵,(2.3)where 𝐴, an 𝑚 by 𝑚 matrix, 𝑋 and 𝐵 are, respectively, given by𝐴_𝑚×𝑚=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣000024120𝑥^𝑛+𝑖…𝜉𝑥^{𝑘−2𝑛+1}000024120𝑥_𝑛+3…𝜉𝑥^{𝑘−2𝑛+3}…0…0…0…0…24120…𝑥_𝑛+𝑘………𝜉𝑥^{𝑘−2𝑛+𝑘}1𝑥_𝑛𝑥^2𝑛𝑥^3𝑛𝑥^4𝑛𝑥^5𝑛…𝑥^𝑛𝑘+21𝑥_𝑛+1𝑥^2𝑛+1𝑥^3𝑛+1𝑥^4𝑛+1𝑥^5𝑛+1…𝑥^𝑘+2𝑛+1……1𝑥……_{𝑛+𝑘−1}………………………………𝑥^{2𝑛+𝑘−1}𝑥^{3𝑛+𝑘−1}𝑥^{4𝑛+𝑘−1}……𝑥^{𝑘+2𝑛+𝑘−1}⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(2.4)𝑋=[𝑎₀𝑎₁𝑎₂…𝑎_𝑘+1𝑎_𝑘+2]^𝑇, 𝐵=[𝑓_𝑛+1𝑓_𝑛+3…𝑓_𝑛+𝑘𝑦_𝑛𝑦_𝑛+1…𝑦_{𝑛+𝑘−1}]^𝑇, 𝜉=(𝑘−1)𝑘(𝑘+1)(𝑘+2),

𝑓_𝑛+𝑖=𝑓(𝑥_𝑛+𝑖,𝑦_𝑛+𝑖,𝑦^𝑛+𝑖,𝑦^{𝑛+𝑖},𝑦^{𝑛+𝑖}),𝑖=0,1,2,…,𝑦_𝑛+𝑖=𝑦(𝑥_𝑛+𝑖), and 𝑇 is the usual matrix transpose.

Using the Gaussian method to solve the matrix (2.3), the values of 𝑎^𝑗𝑠,𝑗=0,1,…,𝑘+2, for 𝑘=5, are obtained as shown in Appendix A. These values of 𝑎^𝑗𝑠 are substituted into (2.1). By using the transformation 𝑥=𝑡ℎ+𝑥_𝑛+4,𝑡∈(0,1] with ℎ being steplength, in the resulting equation after the substitution, an open method with variable coefficients is obtained as𝑦(𝑡)=^𝑘−1_𝑗=0𝛼_𝑗(𝑡)𝑦_𝑛+𝑗+ℎ^4𝑘_𝑗=0𝛽_𝑗(𝑡)𝑓_𝑛+𝑗,𝛽_𝑘≠0.(2.5) Taking 𝑘=5, the coefficients 𝛼_𝑗(𝑥) and 𝛽_𝑗(𝑥) as well as their first, second, and third derivatives in (2.5) are computed and, respectively, obtained as shown in Appendix B.

The values of 𝑡 could be taken in the interval 𝐼=(0,1] in order to obtain a particular discrete scheme. In this work, we take 𝑡=1 in (2.5) to have a zero-stable method and its derivatives as𝑦_𝑛+5=4𝑦_𝑛+4−6𝑦_𝑛+3+4𝑦_𝑛+2−𝑦_𝑛+1+ℎ⁴24𝑓^𝑛+5+22𝑓_𝑛+3+𝑓𝑦^𝑛+1,(2.6)_𝑛+5=1210ℎ853𝑦^𝑛+4−1767𝑦_𝑛+3+1128𝑦_𝑛+2−157𝑦_𝑛+1−57𝑦_𝑛+ℎ⁴481319𝑓^𝑛+5𝑦+20738𝑓_𝑛+3(2+.71)_𝑛+5679𝑓_𝑛+1,=160ℎ²119𝑦^𝑛+4−236𝑦_𝑛+3+54𝑦_𝑛+2+124𝑦_𝑛+1−60𝑦_𝑛+ℎ⁴8159𝑓^𝑛+5𝑦+1526𝑓_𝑛+3+(220.83)_{𝑛+5}𝑓_𝑛+1,=140ℎ³−23𝑦^𝑛+4+132𝑦_𝑛+3−258𝑦_𝑛+2+212𝑦_𝑛+1−63𝑦_𝑛+ℎ⁴12317𝑓^𝑛+5+1364𝑓_𝑛+3+(227.95)𝑓_𝑛+1,𝑝=6 respectively.

The order of the open method (2.6) and its derivatives (2.7), (2.8), (2.9) is the same and is 𝐶_𝑝+2≈−4.31×10⁻². The error constant of (2.6) is found to be 𝛽_𝑘≠0. The method is consistent and zero stable, satisfying the necessary and sufficient conditions for convergence of linear multistep methods [3, 4].

3. Derivation of Starting Values for the Method

Method (2.6) above and its derivatives are open; 𝛽_𝑘=0. This implies that the sample discrete method (2.6) and its derivatives (2.7), (2.8), (2.9) require additional closed starting values, 𝑓_𝑛+𝑖,𝑖=1,2,…,𝑘, for the evaluation of 𝑓_𝑛+𝑖,𝑖=4,…,𝑘. In this work, efforts are made at obtaining the main closed predictors of the same order of accuracy as starting values for 𝑘=5,4. These predictors are obtained by using the same approach described in Section 2. The following consistent and zero-stable closed methods are obtained, for 𝑦(𝑡)=^𝑘−1_𝑗=0𝛼_𝑗(𝑡)𝑦_𝑛+𝑗+^𝑘−1_𝑗=0𝛽_𝑗(𝑡)𝑓_𝑛+𝑗,𝑘=5,4.(3.1),𝑡=1 Taking 𝑘=5 and for 𝑦_𝑛+5=4𝑦_𝑛+4−6𝑦_𝑛+3+4𝑦_𝑛+2−𝑦_𝑛+1+ℎ⁴6𝑓^𝑛+4+4𝑓_𝑛+3+𝑓𝑦^𝑛+2,(3.2)_𝑁+5=1210ℎ898𝑦^𝑛+4−1947𝑦_𝑛+3+1398𝑦_𝑛+2−337𝑦_𝑛+1−12𝑦_𝑛+ℎ⁴12159𝑓^𝑛+4𝑦+132𝑓_𝑛+3+11(53𝑓.3)^𝑛+2_𝑛+5,=115ℎ²38𝑦^𝑛+4−92𝑦_𝑛+3+63𝑦_𝑛+2−2𝑦_𝑛+1−7𝑦_𝑛+ℎ⁴8159𝑓^𝑛+4𝑦+132𝑓_𝑛+3+11(53𝑓.4)^𝑛+2_{𝑛+5},=−110ℎ³11𝑦^𝑛+4−54𝑦_𝑛+3+96𝑦_𝑛+2−74𝑦_𝑛+1+21𝑦_𝑛+ℎ⁴12317𝑓^𝑛+4−124𝑓_𝑛+3+35(93𝑓.5)^𝑛+2,, we have 𝐶_𝑝+21.389×10⁻³ The main predictors (3.2) and its associated derivatives (3.3), (3.4), and (3.5) are each of order 6 and their error constants 6.948×10⁻³ are, respectively, 7.0367×10⁻², 3.269×10⁻¹, 𝑘=4, 𝑦_𝑛+4=4𝑦_𝑛+3−6𝑦_𝑛+2+4𝑦_𝑛+1−𝑦_𝑛+ℎ⁴6(𝑓^𝑛+3+4𝑓_𝑛+2+𝑓_𝑛+1),(3.6).

For 𝑝=6, we have the following: 𝐶_𝑝+2≈1.389×10⁻³ order 𝑦^(1)𝑛+4=16ℎ26𝑦^𝑛+3−57𝑦_𝑛+2+42𝑦_𝑛+1−11𝑦_𝑛+ℎ⁴60185𝑓^𝑛+3+452𝑓_𝑛+2+113𝑓_𝑛+1,(3.7), 𝑝=5;𝐶_𝑝+2≈9.524×10⁻³ order 𝑦^(2)𝑛+4=1ℎ²3𝑦^𝑛+3−8𝑦_𝑛+2+7𝑦_𝑛+1−2𝑦_𝑛+ℎ⁴60449𝑓^𝑛+3+452𝑓_𝑛+2+149𝑓_𝑛+1,(3.8), 𝑝=5;𝐶_𝑝+2≈7.778×10⁻² order 𝑦^(3)𝑛+4=1ℎ³𝑦^𝑛+3−3𝑦_𝑛+2+3𝑦_𝑛+1−𝑦_𝑛+ℎ⁴2455𝑓^𝑛+48𝑓_𝑛+2+13𝑓_𝑛+1,(3.9), 𝑝=5;𝐶_𝑝+2≈0.35 order 𝑘<4, 𝑘.

For 𝑦_𝑛+𝑖,𝑦^(1)𝑛+𝑖,𝑦^(2)𝑛+𝑖,𝑦^(3)𝑛+𝑖,𝑖=1,2,3.

To be able to have sufficient collocation and interpolation points at the grid points that will not produce system of underdetermined equations, the minimum stepnumber 𝑦₀,𝑦⁰⁽¹⁾,𝑦⁰⁽²⁾,𝑦⁰⁽³⁾ required for the derivation of any method for fourth-order problem is 4. Therefore, Taylor series expansion was adopted to evaluate 𝑦^0(𝑟)=𝑦^(𝑟)(𝑥₀),𝑟=0,1,2,3; see Appendix C.

The initial values for the problems are used to evaluate 𝑦⁽⁴⁾+𝑦⁽²⁾=0,0≤𝑥≤𝜋𝑦2,𝑦(0)=0,₍₁₎(0)=−1.172−50Π,𝑦⁽²⁾(0)=1𝑦144−100Π₍₃₎,(0)=1.2Theoretica1l4s4o−lut1i0o0nΠ:𝑦.(𝑥)=(1−𝑥−cos𝑥−1.2sin𝑥)(144−100𝜋).(4.1), where 𝑦𝑦(0⁽⁴⁾)==1𝑦,⁽¹⁾𝑦²₍₁₎−𝑦𝑦^(#)−4𝑥²+𝑒^𝑥1−4𝑥+𝑥²,0≤𝑥≤1,Theoreticalsol(u0t)io=n:1𝑦,(𝑥𝑦)=⁽²⁾(𝑥0₂)=3,𝑦⁽³⁾(0)=1.+𝑒^𝑥.(4.2).

4. Numerical Experience

The accuracy of the proposed method (2.6) has been tested, and the results of the following two initial value problems are shown in the tables in Appendix C below.

Test Problem 1
ℎ=1/32

Test Problem 2
𝑋

The test problems 1 and 2 were solved with method (2.6). The steplength 𝑌 is used for the purpose of comparison with the existing method Kayode [3]. The maximum errors obtained are compared with the method in Kayode [3] as shown in Tables 1 and 2 .

Table 1 

Comparison of errors in Kayode [3] and new method (2.6) for Problem 1.

Table 2 

Comparison of errors in Kayode [3] and new method (2.6) for Problem 2.

5. Conclusion

A predictor-corrector method whose main predictors (3.2) and (3.6) have the same order of accuracy with the corrector (2.6) was formulated. The method was tested to be consistent and zero stable with order six. It is remarkable to note that the main predictors are of the same order 6 as that of the method with comparable error constants. This, to a large extent, reduces the effects that global error could have on the accuracy of the method.

The method was used to solve both linear and nonlinear problems of fourth-order differential equations without reduction to system of first order equations.

The methods (3.2) and (3.6) developed as main predictors are of the same order of accuracy with the corrector (2.6). The method in Kayode [3] has predictors of lower order of accuracy, this accounts for an improvement of method (2.6) over that of [3] as compared in the maximum errors shown in Tables 1 and 2 .

Appendices

A. The Values of 𝛼_𝑗(𝑥)

𝛽_𝑗(𝑥)

B. Coefficients 𝛼₀(𝑡)=11680132𝑡+98𝑡²𝛼₁−126𝑡³−105𝑡⁴−7𝑡⁵+7𝑡⁶+𝑡⁷,(𝑡)=−1420272𝑡+308𝑡²𝛼₂−56𝑡³−105𝑡⁴−7𝑡⁵+7𝑡⁶+𝑡⁷,(𝑡)=1280552𝑡+658𝑡²𝛼₃+14𝑡³−105𝑡⁴−7𝑡⁵+7𝑡⁶+𝑡⁷,(𝑡)=−14201392𝑡+1148𝑡²𝛼₄+84𝑡³−105𝑡⁴−7𝑡⁵+7𝑡⁶+𝑡⁷,(𝑡)=116801680+3212𝑡+1778𝑡²𝛽₁+154𝑡³−105𝑡⁴−7𝑡⁵+7𝑡⁶+𝑡⁷,(𝑡)=120160−240𝑡+284𝑡²𝛽₃+630𝑡³+420𝑡⁴+35𝑡⁵−28𝑡⁶−5𝑡⁷,(𝑡)=1100801824𝑡+3920𝑡²𝛽₅+2814𝑡³+735𝑡⁴−14𝑡⁵−35𝑡⁶−4𝑡⁷,(𝑡)=12016048𝑡+196𝑡²+294𝑡³+210𝑡⁴+77𝑡⁵+14𝑡⁶+𝑡⁷.(B.1) and 𝛼_𝑗(𝑥)

𝛽_𝑗(𝑥) First derivative of 𝛼^0(𝑡)=11680ℎ132+196𝑡−378𝑡²𝛼_1−420𝑡³−35𝑡⁴+42𝑡⁵+7𝑡⁶,(𝑡)=−1420ℎ272+616𝑡−168𝑡²𝛼_2−420𝑡³−35𝑡⁴+42𝑡⁵+7𝑡⁶,(𝑡)=1280ℎ552+1316𝑡+42𝑡²𝛼_3−420𝑡³−35𝑡⁴+42𝑡⁵+7𝑡⁶,(𝑡)=−1420ℎ1392+2296𝑡+252𝑡²𝛼_4−420𝑡³−35𝑡⁴+42𝑡⁵+7𝑡⁶,(𝑡)=11680ℎ3212+3556𝑡+462𝑡²𝛽_1−420𝑡³−35𝑡⁴+42𝑡⁵+7𝑡⁶,(𝑡)=120160ℎ−240+56𝑡+1890𝑡²𝛽_3+1680𝑡³+175𝑡⁴−168𝑡⁵−35𝑡⁶,(𝑡)=110080ℎ1824+7820𝑡+8442𝑡²𝛽_5+2940𝑡³−70𝑡⁴−210𝑡⁵−28𝑡⁶,(𝑡)=120160ℎ48+392𝑡+882𝑡²+840𝑡³+385𝑡⁴+84𝑡⁵+7𝑡⁶.(B.2) and 𝛼_𝑗(𝑥): 𝛽_𝑗(𝑥) Second derivative of 𝛼^0(𝑡)=11680ℎ²196−756𝑡−1260𝑡²𝛼_1−140𝑡³+210𝑡⁴+42𝑡⁵,(𝑡)=−1420ℎ²616−336𝑡−1260𝑡²𝛼_2−140𝑡³+210𝑡⁴+42𝑡⁵,(𝑡)=1280ℎ²1316+84𝑡−1260𝑡²𝛼_3−140𝑡³+210𝑡⁴+42𝑡⁵,(𝑡)=−1420ℎ²2296504𝑡−1260𝑡²𝛼_4−140𝑡³+210𝑡⁴+42𝑡⁵,(𝑡)=11680ℎ²3556924𝑡−1260𝑡²𝛽_1−140𝑡³+210𝑡⁴+42𝑡⁵,(𝑡)=120160ℎ²56+3780𝑡+5040𝑡²𝛽_3+700𝑡³−840𝑡⁴−210𝑡⁵,(𝑡)=110080ℎ²7820+16884𝑡+8820𝑡²𝛽_5−280𝑡³−1050𝑡⁴−168𝑡⁵,(𝑡)=120160ℎ²392+1764𝑡+2520𝑡²+1540𝑡³420𝑡⁴+42𝑡⁵.(B.3) and 𝛼_𝑗(𝑥): 𝛽_𝑗(𝑥) Third derivative of 𝛼^0(𝑡)=11680ℎ³−756−2520𝑡−420𝑡²𝛼_1+840𝑡³+210𝑡⁴,(𝑡)=−1420ℎ³−336−2520𝑡−420𝑡²𝛼_2+840𝑡³+210𝑡⁴,(𝑡)=1280ℎ³84−2520𝑡−420𝑡²𝛼_3+840𝑡³+210𝑡⁴,(𝑡)=−1420ℎ³5044−2520𝑡−420𝑡²𝛼_4+840𝑡³+210𝑡⁴,(𝑡)=11680ℎ³924−2520𝑡−420𝑡²𝛽_1+840𝑡³+210𝑡⁴,(𝑡)=120160ℎ⁴3780+10080𝑡+2100𝑡²𝛽_3−3360𝑡³−1050𝑡⁴,(𝑡)=110080ℎ⁴16884+17640𝑡+2100𝑡²𝛽_5−3360𝑡³−1050𝑡⁴,(𝑡)=120160ℎ⁴1764+5040𝑡+4620𝑡²+1680𝑡³+2100𝑡⁴.(B.4) and 𝑦_𝑛+𝑖≡𝑦𝑥_𝑛+𝑖ℎ≅𝑦𝑥_𝑛+𝑖ℎ𝑦⁽¹⁾𝑥^𝑛+(𝑖ℎ)²2!𝑦⁽²⁾𝑥^𝑛+(𝑖3ℎ!)³𝑦⁽³⁾𝑥^𝑛+(𝑖4ℎ!)⁴𝑓^𝑛+(𝑖ℎ)⁵5!𝑓^𝑛(1)𝑦+…,_(1)𝑛+𝑖≡𝑦⁽¹⁾𝑥^𝑛+𝑖ℎ≅𝑦⁽¹⁾𝑥^𝑛+𝑖ℎ𝑦⁽²⁾𝑥^𝑛+(𝑖ℎ)²2!𝑦⁽³⁾𝑥^𝑛+(𝑖3ℎ!)³𝑓^𝑛+(𝑖ℎ)⁴4!𝑓^𝑛(1)+(𝑖ℎ)⁵5!𝑓^𝑛(2)𝑦+…,_(2)𝑛+𝑖≡𝑦⁽²⁾𝑥^𝑛+𝑖ℎ≅𝑦⁽²⁾𝑥^𝑛+𝑖ℎ𝑦⁽³⁾𝑥^𝑛+(𝑖ℎ)²2!𝑓^𝑛+(𝑖ℎ)³3!𝑓^𝑛(1)+(𝑖ℎ)⁴4!𝑓^𝑛(2)+(𝑖ℎ)⁵5!𝑓^𝑛(3)𝑦+…,_(3)𝑛+𝑖≡𝑦⁽³⁾𝑥^𝑛+𝑖ℎ≅𝑦⁽³⁾𝑥^𝑛+𝑖ℎ𝑓+(𝑖ℎ)²2!𝑓^𝑛(1)+(𝑖ℎ)³3!𝑓^𝑛(2)+(𝑖ℎ)⁴4!𝑓^𝑛(3)+….(C.1): 𝑦⁽⁴⁾(𝑥_𝑛)=𝑓_𝑛,𝑦^𝑛(4+𝑗)=𝑓^𝑛(𝑗),𝑗=0,1,2,…,𝑓^𝑛(0)=𝑓_𝑛

C. Taylor Series

𝑓^𝑛(𝑗)=𝑓^(𝑗)(𝑥_𝑛,𝑦_𝑛,𝑦^𝑛(1),𝑦^𝑛(2),𝑦^𝑛(3)) Noting that in (1.1) 𝑓^𝑛(1),𝑓^𝑛(2), and 𝑓⁽¹⁾=𝑑𝑓𝑑𝑥=𝜕𝜕𝑓𝑥+𝑦⁽¹⁾𝜕𝑓𝜕𝑦+𝑦⁽²⁾𝜕𝑓𝜕𝑦⁽¹⁾+𝑦⁽³⁾𝜕𝑓𝜕𝑦⁽²⁾+𝑓𝜕𝑓𝑓₍₂₎𝜕𝑦⁽³⁾,=𝑑²𝑓𝑑𝑥²=2𝐴𝑦⁽¹⁾+𝐵𝑦⁽²⁾+𝐶𝑦⁽³⁾+𝐷𝑓+𝐸+𝐹,(C.2), the values of 𝐴=𝜕²𝑓𝜕𝑥𝜕𝑦+𝑦⁽²⁾𝜕²𝑓𝜕𝑦𝜕𝑦⁽¹⁾+𝑦⁽³⁾𝜕²𝑓𝜕𝑦𝜕𝑦⁽²⁾+𝑓𝜕²𝑓𝜕𝑦𝜕𝑦⁽³⁾𝐵=𝜕₂,𝑓𝜕𝑥𝜕𝑦+𝑦⁽³⁾𝜕²𝑓𝜕𝑦⁽¹⁾𝜕𝑦⁽²⁾+𝑓𝜕²𝑓𝜕𝑦⁽¹⁾𝜕𝑦⁽³⁾𝐶=𝜕₂,𝑓𝜕𝑥𝜕𝑦⁽²⁾+𝑓𝜕²𝑓𝜕𝑦⁽²⁾𝜕𝑦⁽³⁾𝐷=𝜕₂,𝑓𝜕𝑥𝜕𝑦⁽³⁾𝐸=𝑦₍₂₎,𝜕𝑓𝜕𝑦+𝑦⁽³⁾𝜕𝑓𝜕𝑦⁽¹⁾+𝑓𝜕𝑓𝜕𝑦⁽²⁾+𝑓^𝜕𝑓𝜕𝑦⁽³⁾𝐹=𝜕₂,𝑓𝜕𝑥²+𝑦⁽¹⁾²𝜕²𝑓𝜕𝑦²+𝑦⁽²⁾²𝜕²𝑓(𝜕𝑦⁽¹⁾)²+𝑦⁽³⁾²𝜕²𝑓(𝜕𝑦⁽²⁾)+(𝑓)²𝜕²𝑓𝜕𝑦⁽³⁾)².(C.3) are obtained by partial derivative technique as where