In this paper, we present a method of computer calculation of Green functions in the form of generalized power series for third-order linear differential equations admitting regular singularities. For specific boundary-value problems, we construct Green functions by using the software propose

Belyaeva, I. N.

Chekanov, N. A.

Chekanova, N. N.

Krasovskaya, L. V.

DSpace at Belgorod State University

Сomputer calculation of green functions for third-order ordinary differential equations

Journal of Mathematical Sciences, Vol. 259, No. 3, December, 2021COMPUTER CALCULATION OF GREEN FUNCTIONSFOR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONSI. N. Belyaeva, N. A. Chekanov,L. V. Krasovskaya, and N. N. Chekanova UDC 519.624.3Abstract. In this paper, we present a method of computer calculation of Green functions in the formof generalized power series for third-order linear differential equations admitting regular singularities.For specific boundary-value problems, we construct Green functions by using the software proposed.Keywords and phrases: ordinary differential equation, boundary-value problem, Green function,symbol-numerical computation, computer simulation.AMS Subject Classification: 34B271. Introduction. Green functions are very useful in solving boundary-value problems; for example,they allow one(1) to search for solutions of inhomogeneous ordinary differential equations,(2) to calculate eigenvalues and eigenfunctions for a given differential operator,(3) to represent of an original boundary-value problem in the form of the Fredholm integral equation,etc. (see [2, 3, 5, 6]).Construction of Green functions is a very actual, but quite a difficult problem.For constructing Green function, it is necessary to find all linearly independent solutions of theoriginal boundary-value problem. Thus, an important problem of integrating linear ordinary differ-ential equations arises; having solved this problem, one can then calculate the Green function of theboundary-value problem for an ordinary differential equation by computer methods. In this work, wedescribe solution of third-order equations by using the computer algebra system “Maple.”2. Calculation of Green functions for third-order ordinary differential equations. Con-sider the following third-order differential equation:p3(x)y′′′ + p2(x)y′′ + p1(x)y′ + p0(x)y = 0; (1)here the coefficients p3(x), p2(x), p1(x), and p0(x) are twice continuously differentiable functions onthe segment [a, b]. For Eq. (1), we impose the boundary conditions⎧⎪⎨⎪⎩α1,0y(a) + α1,1y′(a) + α1,2y′′(a) + α1,3y′′′(a) + β1,0y(b) + β1,1y′(b) + β1,2y′′(b) + β1,3y′′′(b) = 0,α2,0y(a) + α2,1y′(a) + α2,2y′′(a) + α2,3y′′′(a) + β2,0y(b) + β2,1y′(b) + β2,2y′′(b) + β2,3y′′′(b) = 0,α3,0y(a) + α3,1y′(a) + α3,2y′′(a) + α3,3y′′′(a) + β3,0y(b) + β3,1y′(b) + β3,2y′′(b) + β3,3y′′′(b) = 0,(2)where αi,k, βi,k, i = 1, 2, 3, k = 0, 1, 2, 3, are coefficients such that∑i∑k(α2i,k + β2i,k) = 0.Assume that the problem (1) with the boundary conditions (2) has only the trivial solution in theclass of twice continuously differentiable functions. However, if we reject the continuity condition forTranslated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. TematicheskieObzory, Vol. 165, Proceedings of the IV International Scientific Conference “Actual Problems of Applied Math-ematics,” Kabardino-Balkar Republic, Nalchik, Elbrus Region, May 22–26, 2018. Part I, 2019.1072–3374/21/2593–0265 c© 2021 Springer Science+Business Media, LLC 265DOI 10.1007/s10958-021-05615-9the second derivative at a point x = ξ, ξ ∈ [a, b], then there exists a nonzero solution G(x, ξ) calledthe Green function.To construct the Green function for the boundary-value problem (1) with the conditions (2), wefirst must solve the Cauchy problem at the point x0. We search for three linearly independent solutionsof Eq. (1) in the form (see [1])y1(x) = 1 +N∑k=3c(1)k (x− x0)k,y2(x) = (x− x0) +N∑k=3c(2)k (x− x0)k,y3(x) = (x− x0)2/2 +N∑k=3c(3)k (x− x0)k,(3)where c(1)k , c(2)k , and c(3)k are numerical coefficients. Then the Green function has the form:G(x, ξ) ={GL(x, ξ), a ≤ x ≤ ξ ≤ b,GR(x, ξ), a ≤ ξ ≤ x ≤ b,GL(x, ξ) =3∑k=1[Ak(ξ) +Bk(ξ)] yk(x),GR(x, ξ) =3∑k=1[Ak(ξ)−Bk(ξ)] yk(x).(4)Using the basic properties of Green functions, we find the unknown functional coefficients Ak(ξ)and Bk(ξ).1. The Green function and its first derivative are continuous at the point x = ξ.2. At the point x = ξ, the second derivative of the Green function has a jump discontinuity:∂2G(x, ξ)∂x2∣∣∣x=ξ+0−∂2G(x, ξ)∂x2∣∣∣x=ξ−0=1p3(ξ). (5)3. The Green function satisfies the differential equation (1).4. The Green function satisfies the boundary conditions (2).A theorem from the theory of differential equations (see, e.g., [4, 7]) states that if the boundary-value problem (1) with the conditions (2) has only trivial solution y(x) ≡ 0, then there exists a uniqueGreen function G(x, ξ).Note that, since we search for a solution of the Cauchy problem if the power-series form, the Greenfunction G(x, ξ) will be also obtained in this form.From the properties 1 and 2 of the Green function we obtain the following system of algebraicequations:3∑k=1Bk(ξ) · yk(ξ) = 0,3∑k=1Bk(ξ) · y′k(ξ) = 0,3∑k=1Bk(ξ) · y′′k(ξ) = −12p3(ξ). (6)Solving the system (6), we find the coefficients Bk(ξ), k = 1, 2, 3. Note that the determinant of thissystem is the Wronskian of the linearly independent solutions y1, y2, y3:W (ξ) ≡ W [y1, y2, y3] =∣∣∣∣∣∣y1 y2 y3y′1 y′2 y′3y′′1 y′′2 y′′3∣∣∣∣∣∣= 0. (7)266Therefore, the system of algebraic equations (6) possesses a unique solution Bk(ξ), k = 1, 2, 3. Fur-ther, we calculate the functional coefficients Ak(ξ), k = 1, 2, 3. Taking into account the boundaryconditions (2), we arrive at the following system of three equations:A1(ξ) [α1,0y1(a) + α1,1y′1(a) + α1,2y′′1 (a) + α1,3y′′′1 (a) + β1,0y1(b) + β1,1y′1(b) + β1,2y′′1 (b) + β1,3y′′′1 (b)] ++A2(ξ) [α1,0y1(a) + α1,1y′1(a) + α1,2y′′1 (a) + α1,3y′′′1 (a) + β1,0y1(b) + β1,1y′1(b) + β1,2y′′1 (b) + β1,3y′′′1 (b)] ++A3(ξ) [α1,0y1(a) + α1,1y′1(a) + α1,2y′′1 (a) + α1,3y′′′1 (a) + β1,0y1(b) + β1,1y′1(b) + β1,2y′′1 (b) + β1,3y′′′1 (b)] == B1(ξ) [α1,0y1(a) + α1,1y′1(a) + α1,2y′′1 (a) + α1,3y′′′1 (a)− β1,0y1(b)− β1,1y′1(b)− β1,2y′′1 (b)− β1,3y′′′1 (b)] ++B2(ξ) [α1,0y1(a) + α1,1y′1(a) + α1,2y′′1 (a) + α1,3y′′′1 (a)− β1,0y1(b)− β1,1y′1(b)− β1,2y′′1 (b)− β1,3y′′′1 (b)] ++B3(ξ) [α1,0y1(a) + α1,1y′1(a) + α1,2y′′1 (a) + α1,3y′′′1 (a)− β1,0y1(b)− β1,1y′1(b)− β1,2y′′1 (b)− β1,3y′′′1 (b)] , (8)A1(ξ) [α2,0y1(a) + α2,1y′1(a) + α2,2y′′1 (a) + α2,3y′′′1 (a) + β2,0y1(b) + β2,1y′1(b) + β2,2y′′1 (b) + β2,3y′′′1 (b)] ++A2(ξ) [α2,0y1(a) + α2,1y′1(a) + α2,2y′′1 (a) + α2,3y′′′1 (a) + β1,0y1(b) + β2,1y′1(b) + β2,2y′′1 (b) + β2,3y′′′1 (b)] ++A3(ξ) [α2,0y1(a) + α2,1y′1(a) + α2,2y′′1 (a) + α2,3y′′′1 (a) + β2,0y1(b) + β2,1y′1(b) + β2,2y′′1 (b) + β2,3y′′′1 (b)] == B1(ξ) [α2,0y1(a) + α2,1y′1(a) + α2,2y′′1 (a) + α2,3y′′′1 (a)− β2,0y1(b)− β2,1y′1(b)− β2,2y′′1 (b)− β2,3y′′′1 (b)] ++B2(ξ) [α2,0y1(a) + α2,1y′1(a) + α2,2y′′1 (a) + α2,3y′′′1 (a)− β2,0y1(b)− β2,1y′1(b)− β2,2y′′1 (b)− β2,3y′′′1 (b)] ++B3(ξ) [α2,0y1(a) + α2,1y′1(a) + α2,2y′′1 (a) + α2,3y′′′1 (a)− β2,0y1(b)− β2,1y′1(b)− β2,2y′′1 (b)− β2,3y′′′1 (b)] , (9)A1(ξ) [α3,0y1(a) + α3,1y′1(a) + α3,2y′′1 (a) + α3,3y′′′1 (a) + β3,0y1(b) + β3,1y′1(b) + β3,2y′′1 (b) + β3,3y′′′1 (b)] ++A2(ξ) [α3,0y1(a) + α3,1y′1(a) + α3,2y′′1 (a) + α3,3y′′′1 (a) + β3,0y1(b) + β3,1y′1(b) + β3,2y′′1 (b) + β3,3y′′′1 (b)] ++A3(ξ) [α3,0y1(a) + α3,1y′1(a) + α3,2y′′1 (a) + α3,3y′′′1 (a) + β3,0y1(b) + β3,1y′1(b) + β3,2y′′1 (b) + β3,3y′′′1 (b)] == B1(ξ) [α3,0y1(a) + α3,1y′1(a) + α3,2y′′1 (a) + α3,3y′′′1 (a)− β3,0y1(b)− β3,1y′1(b)− β3,2y′′1 (b)− β3,3y′′′1 (b)] ++B2(ξ) [α3,0y1(a) + α3,1y′1(a) + α3,2y′′1 (a) + α3,3y′′′1 (a)− β3,0y1(b)− β3,1y′1(b)− β3,2y′′1 (b)− β3,3y′′′1 (b)] ++B3(ξ) [α3,0y1(a) + α3,1y′1(a) + α3,2y′′1 (a) + α3,3y′′′1 (a)− β3,0y1(b)− β3,1y′1(b)− β3,2y′′1 (b)− β3,3y′′′1 (b)] . (10)Substituting the functional coefficients Ak(ξ) and Bk(ξ), k = 1, 2, 3, into the formula (4), we obtainthe Green function in the analytic form.3. Algorithm of the search for Green functions for third-order ordinary differentialequations.Input:• p3(x), p2(x), p1(x), p0(x) are functional coefficients in the third-order ordinary differential equa-tion (1),• N is the number of terms of the power series (3),• a and b are the boundary points of the segment on which the Green function GL(x, ξ) will beconstructed,• α1,0, α1,1, α1,2, α1,3, α2,0, α2,1, α2,2, α2,3, α3,0, α3,1, α3,2, α3,3, β1,0, β1,1, β1,2, β1,3, β2,0, β2,1,β2,2, β2,3, β3,0, β3,1, β3,2, β3,3 are numerical coefficients for the boundary-value problem in theconditions (2).Output:• y1(x), y2(x), y3(x) are linearly independent solutions of the given third-order ordinary differen-tial equation (1),• GL(x, ξ) is the Green function on the segment [a, b], where a ≤ x ≤ ξ ≤ b,• GR(x, ξ) is the Green function on the segment [a, b], where a ≤ ξ ≤ x ≤ b.Description of the algorithm:2671. Construction of three linearly independent solutions of the third-order differential equation inthe generalized power-series form (see [1]).2. Verification of the solutions constructed.3. Solution of the system of algebraic equations (6) for the functional coefficients Bk(ξ), k = 1, 2, 3.4. Verification of the functional coefficients Bk(ξ), k = 1, 2, 3.5. Solution of the system of algebraic equations (8), (9), (10) for the coefficients Ak(ξ), k = 1, 2, 3.6. Verification of the coefficients Ak(ξ), k = 1, 2, 3.7. Construction of the Green functions GL(x, ξ) under the condition a ≤ x ≤ ξ ≤ b by the for-mula (4).8. Construction of the Green function GR(x, ξ) under the condition a ≤ ξ ≤ x ≤ b by the formula 4.9. Verification of the four basic properties of the Green function G(x, ξ).4. Computer simulation of Green functions. The software developed was tested on the differ-ential equationy′′′ − 6y′′ + 11y′ − 6y = 0 (11)with the boundary conditionsy(0) = 0, y(1) = 0, y′′(0) = 0. (12)First, we construct the Green function for this boundary-value problem without using a computer.The fundamental system of solutions of Eq. (11) with the conditions (12) has the formy1 = ex, y2 = e2x, y2 = e3x. (13)The function G(x, ξ) can be constructed from the solutions (13) by the algorithm proposed above.Taking into account the basic properties 1 and 2 of the Green function, we obtain the system⎧⎪⎪⎪⎨⎪⎪⎪⎩B1(ξ) · y1(ξ) +B2(ξ) · y2(ξ) +B3(ξ) · y3(ξ) = 0,B1(ξ) · y′1(ξ) +B2(ξ) · y′2(ξ) +B3(ξ) · y′3(ξ) = 0,B1(ξ) · y′′1 (ξ) +B2(ξ) · y′′2(ξ) +B3(ξ) · y′′3(ξ) = −12p3(ξ).(14)Since p3(ξ) = 0, the system (14) is always solvable and has a unique solution. Thus, the principaldeterminant of the system (14) is the Wronskian W [y1, y2, y3] = 0. Solving the system (14), we findthe coefficients Bk(ξ), k = 1, 2, 3:B1(ξ) = −14e−ξ, B2(ξ) =12e−2ξ , B3(ξ) = −14e−3ξ . (15)Further, taking into account the boundary conditions (12), we obtain the system of equations⎧⎪⎨⎪⎩A1(ξ) +A2(ξ) +A3(ξ) = −B1(ξ)−B2(ξ)−B3(ξ),A1(ξ) + 4A2(ξ) + 9A3(ξ) = −B1(ξ)− 4B2(ξ)− 9B3(ξ),A1(ξ) + eA2(ξ) + e2A3(ξ) = B1(ξ) + eB2(ξ) + e2B3(ξ).(16)Solving this system, we obtain the coefficients Ak(ξ), k = 1, 2, 3:A1(ξ) = − 14(3e2 − 8e+ 5)(− 20e1+ξ + 5e2ξ + 10e2 + 8e1−2ξ − 3e2+2ξ)e−3ξ,A2(ξ) =12(3e2 − 8e+ 5)(− 8e1+ξ + 8e2ξ + 8e2 − 5eξ − 3e2+ξ)e−3ξ,A3(ξ) = − 14(3e2 − 8e+ 5)(− 12e1+ξ − 5 + 6e2ξ + 3e2 + 8e)e−3ξ.(17)268Substituting the functions Bk(ξ) and Ak(ξ), k = 1, 2, 3, into (4), we obtain the exact Green functionfor the problem (11) with the conditions (12) in the formG(x, ξ) ={GL(x, ξ), 0 ≤ x ≤ ξ ≤ 1,GR(x, ξ), 0 ≤ x ≤ ξ ≤ 1, (18)GL(x, ξ) =ex(−5/2e−ξ + 5e1−2ξ − 5/2e2−3ξ)3e2 − 8e+ 5 +e2x(4e−ξ − 8e1−2ξ + 4e2−3ξ)3e2 − 8e+ 5 ++e3x(−3/2e−ξ + 3e1−2ξ − 3/2e2−3ξ)3e2 − 8e+ 5 ,GR(x, ξ) =ex(3/2e2−ξ − 4e1−ξ + 5e1−2ξ − 5/2e2−3ξ)3e2 − 8e+ 5 +e2x(4e−ξ − 3e2−2ξ − 5e−2ξ + 42−3ξ)3e2 − 8e+ 5 ++e3x(−3/2e−ξ + 3e1−2ξ + 5/2e−3ξ − 4e1−3ξ)3e2 − 8e+ 5 .Using the software for the Maple computer algebra system, we computed an approximate Greenfunction (with N = 6) for the boundary-value problem (11) under the conditions (12):G(x, ξ) ={G6L(x, ξ), 0 ≤ x ≤ ξ ≤ 1,G6R(x, ξ), 0 ≤ ξ ≤ x ≤ 1,(19)G6L(x, ξ) =1966335489x− 8248935489xξ +30816770978xξ2 − 216293212934x3 − 1059601212934xξ3 +907379212934x3ξ − 9831570978x4++3462703851736xξ4 − 3389837425868x3ξ2 +41244570978x4ξ − 46994574258680x5 − 109696094258680xξ5 +116556111277604x3ξ3−− 1540835141956x4ξ2 +197148714258680x5ξ − 137641212934x6 +4678305112776040xξ6 − 380897335110416x3ξ4++5298005425868x4ξ3 − 736519138517360x5ξ2 +577423212934x6ξ +12066569925552080x3ξ5 − 173135151703472x4ξ4++25324463925552080x5ξ3 − 2157169425868x6ξ2 − 51461356176656240x3ξ6 +109696091703472x2ξ5 − 827586017102208320x5ξ4++74172071277604x6ξ3 − 467830515110416x4ξ6 +2621736551511041600x5ξ5 − 242389215110416x6ξ4 − 111811491891533124800x5ξ6++7678726325552080x6ξ5 − 32748135776656240x6ξ6,G6R(x, ξ) =1966335489x+12ξ2 − 11797835489xξ +12x2 − ξ3 + 52110170978xξ2 − 3x2ξ − 3359212934x3 − 2524ξ4++32447135489xξ3 +254x2ξ2 +335935489x3ξ − 292555851736x4 − 34ξ5 +6656713851736xξ4 − 152x2ξ3−− 195827425868x3ξ2 +292555141956x4ξ − 15054474258680x5 +301720ξ6 − 3608633709780xξ5 +30148x2ξ4++5407970978x3ξ3 − 78078311703472x4ξ2 +1505447709780x5ξ − 583473125552080x6 +319621196388020xξ6−− 16140x2ξ5 − 38073595110416x3ξ4 +1627427283912x4ξ3 − 393695398517360x5ξ2 − 58347314258680x6ξ +605144x2ξ6++22185794258680x3ξ5 − 10040795520441664x4ξ4 +81050231419560x5ξ3 − 3030121110220832x6ξ2 − 2699470176656240x3ξ6.Comparison of the approximate Green function (19) constructed by using the method proposed abovewith the exact Green function calculated by the formula (18) shows that the approximate Green269function Gcalc constructed by N = 13 terms of series differs from the exact Green function Gtheor onlyby 1.3%. Increasing the number of terms leads to decreasing the error; for example, if N = 16, thenthe relative error becomes 0.003%.We also note that the error depends on ξ. Calculations with N = 13 show that the absolute errorΔ = |Gtheor −Gcalc| for ξ = 0.25 is equal to Δ = 8 · 10−7, whereas for ξ = 0.5 we have Δ = 1.5 · 10−6and for ξ = 0.75 we have Δ = 0.00015. The relative error ε = Δ/Gtheor · 100% is equal to ε = 0.038%for ξ = 0.25. If ξ = 0.5, then ε = 1.3%; if ξ = 0.75, then ε = 3%.We also calculate the absolute error for N = 16. In this case, we have Δ = 6 · 10−9 for ξ = 0.25,Δ = 8 ·10−9 for ξ = 0.5, and Δ = 6 ·10−7 for ξ = 0.75. The relative error is ε = 4 ·10−5% for ξ = 0.25,ε = 0.003% for ξ = 0.5, and ε = 0.006% for ξ = 0.75.Consider the differential equationy′′′ = 0 (20)with the boundary conditionsy(0) = 0, y(1) = 0, y′(0) = y′(1). (21)Using the software developed, we obtain the Green functionG(x, ξ) ={GL(x, ξ), 0 ≤ x ≤ ξ ≤ 1,GR(x, ξ), 0 ≤ ξ ≤ x ≤ 1, (22)whereGL =12(x2ξ − xξ2 − x2 + xξ), GR = 12(x2ξ − xξ2 − xξ + ξ2).The expression (22) coincides with the exact formula; it is anti-self-adjoint (see [5, 6]).Now we consider the same differential equation (20) with the other boundary conditions:y(0) = 0, y(1) = 0, y′(0) + y′(1) = 0. (23)This equation has a nontrivial solution; therefore, the Green function G(x, ξ) does nor exist.Further, for the differential equation (20) with the boundary conditionsy(0) = 0, y′(1) = 0, y′(0) = y(1), (24)we obtain the following Green function:G(x, ξ) ={GL(x, ξ), 0 ≤ x ≤ ξ ≤ 1,GR(x, ξ), 0 ≤ ξ ≤ x ≤ 1, (25)whereGL = −12x2ξ2 + x2ξ + xξ2 − 12x2 − xξ, GR = −12x2ξ2 + x2ξ + xξ2 − 2xξ + 12ξ2.Note that in this case, the Green function G(x, ξ) defined by the formula (25) is not anti-self-adjoint.The boundary-value problems discussed above demonstrate the importance of boundary conditions.Consider the differential equationy′′′ − x2y′′ + (a+ b− 1)xy′ − aby = 0 (26)where a and b are parameters, with the boundary conditionsy(0) = 0, y(1) = 0, y′(0) = y′(1). (27)270Using the software developed, we construct the Green function in the power-series form with N = 5.For simplicity, we set a = b = 1; then the Green function has the formG(x, ξ) ={GL(x, ξ), 0 ≤ x ≤ ξ ≤ 1,GR(x, ξ), 0 ≤ ξ ≤ x ≤ 1, (28)GL = − 1120x5 +13x2ξ3 − 24125xξ4 − 325x2ξ2 +1225x2ξ − 173375xξ2 − 12x2 +64125xξ,GR =112x3ξ2 +1831000xξ4 − 215ξ5 − 325x2ξ2 +1225x2ξ − 173375xξ2 − 61125xξ +12ξ2.REFERENCES1. I. N. Belyaeva and N. A. Chekanov, Symbolic-numerical integration of a third-order linear differ-ential equation in the form of generalized series, Certificate of state registration of a computerprogram No. 2010614257 (2010).2. E. L. Ince, Ordinary Differential Equations, Dover, New York (1944).3. E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen. I. Gewöhnliche Differential-gleichungen, B. G. Teubner, Stuttgart (1977).4. M. L. Krasnov, A. I. Kiselev, G. I. Makarenko, Integral Equations [in Russian], Nauka, Moscow(1976).5. G. Sansone, Equazioni Differenziali nel Campo Reale. Vol. 1, Bologna (1948).6. G. Sansone, Equazioni Differenziali nel Campo Reale. Vol. 2, Bologna (1949).7. F. G. Tricomi, Differential Equations, Blackie & Son, London (1961).I. N. BelyaevaBelgorod National Research UniversityE-mail: ibelyaeva@bsu.edu.ruN. A. ChekanovBelgorod National Research UniversityE-mail: Chekanov@bsu.edu.ruL. V. KrasovskayaBelgorod National Research UniversityE-mail: krasovskaya@bsu.edu.ruN. N. ChekanovaBanking University, Kharkiv, UkraineE-mail: chekanova76@list.ru271

Computer calculation of green functions for third-order ordinary differential equations

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Сomputer calculation of green functions for third-order ordinary differential equations

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