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Finite Difference Method for Solving Second-Order Boundary Value Problems With

High-Order Accuracy

Nguyen Dinh Dung∗, Vu Vinh Quang
Thai Nguyen University of Information and Communication Technology, Thai Nguyen, Vietnam

∗Correspondence: nddung@ictu.edu.vn

Abstract. When researching and solving practical problems in continuous environments, throughmodeling methods, the vast majority of problems lead to models described by equations containingdifferential operators. In case the problem model is not complicated, we often obtain simple partialdifferential equations, then the solution of the problem can be obtained directly through analyti-cal methods. Most complex problems, through the method of approximating differential operatorsby difference operators, from which differential problems are approximated by corresponding differ-ence schemes and approximate solutions will be obtained. is achieved through solving systems ofdifference equations based on the tools of electronic computers. Then, building difference schemesto approximate the differential problem with high-order accuracy will play an important role in theaccuracy of the obtained approximate solution. In this paper, we propose two difference schemes withhigh-order accuracy to solve second-order differential problems with Dirichlet and mixed boundaryconditions. Theoretical results and experimental calculations have confirmed the accuracy of theproposed schemes.

1. Introduction
In this paper, we consider the second-order boundary problem with mixed boundary conditions

u′′(x) = f (x), x ∈ (a, b)

c0u(a)− c1u′(a) = C

d0u(b) + d1u
′(b) = D

c0, c1, d0, d1 ≥ 0

(1)
In some cases when the function f (x) is a polynomial function, trigonometric function, exponentialfunction or product of the above functional forms, the solution to problem (1) can be found byanalytical methods or using Using Green’s function method, we must find the approximate solutionof problem (1) by numerical methods using electronic computers.In order to find approximate solutions using numerical methods, we need to build systems ofdifference equations that approximate the differential problem and then find numerical solutions
Received: 11 Feb 2024.
Key words and phrases. Derivative; Grid space; Mesh function; partial differential equations; Difference schemes.1

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https://doi.org/10.28924/ada/ma.4.10


Eur. J. Math. Anal. 10.28924/ada/ma.4.10 2using algebraic methods. Then the accuracy order of approximating differential operators usingdifference operators will determine the accuracy order of the obtained approximated solution.In [1], a method for building a difference scheme for problem (1) was introduced by approximatingthe first and second derivatives of the function u(x) with accuracy O(h), where h is the grid stepin grid space, in [2] a method of building a difference scheme with accuracy O(h2). In addition,some authors [3]- [5] have proposed difference methods with order of accuracy from O(h4) to O(h8).However, these methods lead to very complicated systems of difference equations that requiringthe use of intermediate grid points. This makes solving these systems of difference equationsvery complicated. Therefore, it is required to build a difference scheme with order of accuracy forproblem (1) so that the resulting system of difference equations is simple and can be solved withan algorithm of linear complexity.Studying methods to approximate the derivative value at a point of a function is an importantresearch direction that mathematicians are especially interested in. In the literature on numericalmethods [1], formulas for approximating the first and second derivatives of the function u(x) withaccuracy order O(h2) have been given, which uses the value of the function at 3 neighboring pointsof point x and h is denoted as the grid step on the grid space. Based on the interpolation polynomialapproximation method, in the document [6], the m-order derivative approximation formulas withaccuracy order O(h5−m) are given, which uses value of the function at 5 neighboring points ofpoint x . Recently, in the document [7], a set of formulas for approximating derivatives of allorder with accuracy order O(hn−m) has been introduced, which uses the value of the function at
n + 1 neighboring points of point x . Based on the published formulas, numerical solutions fordifferential problems with nonlinear differential equations of all order have been improved. In [8]-[12], published algorithms for numerically solving 3rd and 4th order nonlinear differential equationswith accuracy order O(h6). However, the published results of the above derivative approximationformulas are all results obtained by the direct method, results obtained by direct calculation ofanalytical expressions, not general. Thus, to improve the accuracy of derivative approximation, itis necessary to build a general algorithm to provide formulas for approximating derivatives of anydegree with arbitrary order of accuracy based on the support of the computer.The main content of the article presents research results based on the Taylo expansion formula [1]and the algorithm for determining numerical derivatives with high order accuracy, thereby proposinga method to build two difference scheme for problem (1) in the case of Dirichlet boundary conditionsand mixed boundary conditions with arbitrary high-order precision. Difference schemes are verysimple and can be solved with the algorithm has O(N) complexity where N is the number of pointsin the grid space. The experimental results we calculated confirm the accuracy of the proposedschemes.

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Eur. J. Math. Anal. 10.28924/ada/ma.4.10 3The structure of the article consists of four parts: In Section 1, we introduce some publishedresults on based on the difference method. Section 2, we present the theoretical basis of thederivative approximation method based on the Taylor formula, then we propose a method to build adifference scheme with high-order accuracy based on the high-order derivative approximation andevaluate schema accuracy. Section 3, we present some experimental calculation results to evaluatethe accuracy of the proposed algorithm. Finally, there are conclusions and references.
2. Proposed method

2.1. Derivative approximation formulas

Consider the function u(x) ∈ Cn+1[a, b], expanding the Taylor series [3], we have the formula
u(x + ∆x) = u(x) + u′(x)∆x +

u′′(x)

2!
(∆x)2 + ...+

u(n)(x)

n!
(∆x)n + Rn(x) (2)

where Rn(x) = u(n+1)(θ)
(n+1)! (∆x)n+1, θ ∈ (x, x + ∆x) Since formula (2), we can approximate thederivative of the function u(x) in the neighborhood x as follows:

u(x + ∆x) ≈ u(x) + u′(x)∆x +
u′′(x)

2!
(∆x)2 + ...+

u(n)(x)

n!
(∆x)n (3)

Divide the interval [a, b] by (n+ 1) grid points xi = a+ ih, i = 0, 1, ..., n, grid step h = b−a
n , sinceformula (3), we get the following formulas

u(xi + h) = u(xi) + u′(xi)h +
u′′(xi)

2!
h2 + ...+

u(n)(xi)

n!
hn +O(hn+1), i = 0, n − 1 (4)

u(xi − h) = u(xi)− u′(xi)h +
u′′(xi)

2!
h2 + ...+ (−1)n

u(n)(xi)

n!
hn +O(hn+1), i = 0, n − 1 (5)

Using (4), we obtain
u′(xi) =

u(xi + h)− u(xi)

h
−
u′′(xi)

2!
h − ...−

u(n)(xi)

n!
hn−1 +O(hn) (6)

Formula (6) is a formula that supports approximating the first derivative on a uniform grid witherror O(hn).Since (4) and (5), we have
u′′(xi) =

u(xi + h)− 2u(xi) + u(xi − h)

h2
−
u(4)(xi)

4!
h2 − ...−

u(2m)(xi)

(2m)!
h2m−2 +O(hn−1) (7)

Let v(x) be a function defined on [a, b], ωh be the grid space with grid step h = (b − a)/N ,
xi = a + ih, (i = 0, 1, 2, . . . , N), V = (V (x0), V (x1), . . . , V (xN)) is the approximated value of the
Vd on ωh, where Vd = (v(x0), v(x1), . . . , v(xN)).

Definition 1: On grid ωh, a method for approximating the function v(x) that is said to have
n-order accuracy if ‖V − Vd‖ωh = O(hn).

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Eur. J. Math. Anal. 10.28924/ada/ma.4.10 4Since (3), (6), (7), we can build derivative approximation formulas with (n+1)- order accuracy forthe grid function, and n-order accuracy for first-order derivative, (n-1)- order accuracy for second-order derivative.
Lemma 1: If the accuracy of the second derivative approximation is of order (n − 2), then the

accuracy of the function approximation will be of order n and vice versa.Let
Ti(x) = (x − z0)(x − z1)...(x − zi−1)(x − zi+1)...(x − zn),

Mi(x) = (zi − z0)(zi − z1)...(zi − zi−1)(zi − zi+1)...(zi − zn).Let B(n(m)) be the coefficient matrix in the m- order derivative approximation formula using
(n + 1) neighboring points, then the grid derivative of orders will be determined by the formula:

u(m)(xi) = 1
hm

n∑
k=0

B
(m)
n (i , k)uk , i < n/2

u(m)(xi) = 1
hm

n∑
k=0

B
(m)
n (i , k)ui+k−n/2, n/2 ≤ i ≤ N − n/2

u(m)(xi) = 1
hm

n∑
k=0

B
(m)
n (i , k)uN−n+k , i > N − n/2

(8)

where,L(m)i (x) = m!
T
(m)
i (x)

Mi
; T (m)i (x) = (x − zm)(x − zm+1)...(x − zn) + (x − z0)(x − zm+1)...(x −

zn) + (x − z0)(x − z1)(x − zm+2)...(x − zn) + ...+ (x − z0)(x − z1)(x − z2)...(x − zn−m).
Theorem 1: The accuracy of approximating the grid derivative of order m using (n + 1) neigh-

boring points on the regular grid is of order (n −m + 1).Formulas (6), (7), (8) will be used to propose the difference scheme given in section 2.2
2.2. Difference scheme with high order accuracyConsider the differential problem

Lnu(x) = f (x), Bau(a) = Ga, Bbu(b) = Gb, x ∈ [a, b], (9)
where Ln is the linear differential operator, Ba, Bb are the boundary condition operators.Let

ΛnU = Φ, ΛaU = Φa, ΛbU = Φb (10)be the difference scheme for the differential problem (9), where Λn is the linear difference operator,
Λa, Λb are the boundary condition difference operators. Let U be the grid function determinedbased on the difference scheme, let Ud be the value of u(x) on the grid ωh. Below we give twodifference schemes corresponding to two differential problems.
2.2.1. Differential scheme 1Consider the second-order boundary problem with dirichlet boundary conditions{

u′′(x) = f (x), x ∈ (a, b)

u(a) = C; u(b) = D.
(11)

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Eur. J. Math. Anal. 10.28924/ada/ma.4.10 5Considering the grid space ωh, let Ui = u(xi), U = (U0, U1, . . . , UN) be the grid function, F =

(F0, F1, . . . , FN) is the right-hand grid function, D(m)F is the m-order derivative of the grid function
F . Since f (x) satisfies the differential equation (11), we have f (xi) = u′′(xi), i = 0, 1, 2, . . . , N .Using (7) we obtain the approximated formula
u′′(xi) =

ui−1 − 2ui + ui+1
h2

−
h2

4!
D(2)Fi −

h4

6!
D(4)Fi − ...−

h2m−2

(2m)!
D(2m−2)Fi +O(hn−1) (12)

Set Φi = h2Fi + h4

4!D
(2)Fi + h6

6!D
(4)Fi + ...+ h2m

(2m)!D
(2m−2)Fi , 1 ≤ i ≤ N − 1, we obtain the systemof difference equations {

Ui−1 − 2Ui + Ui+1 = Φi , i = 1, 2, ..., N − 1

U0 = C; UN = D
(13)

(13) is is called a three-diagonal system because the matrix of the system has a three-diagonal form.Solving system (13) is performed using the following pursuit algorithm. In the case of constructingscheme (13), we used formula (7) which is used to approximate the 2nd derivative with (n−2)-orderaccuracy, therefore, since Lemma 1, it is implies solution of system (13) is the approximation of thefunction with n-order accuracy. Since there we have the following theorem:
Theorem 2: The system of difference equations (13) is stable and its solution approximates to

solution of problem (11) with n-order accuracy.The solution of the system is obtained from the pursuit algorithm which has complexity O(N).
2.2.2. Differential scheme 2Consider the second-order boundary problem with mixed boundary conditions{

u′′(x) = f (x), x ∈ (a, b)

c0u(a)− c1u′(a) = C; d0u(b) + d1u
′(b) = D

(14)
Consider the grid space ωh, set Ui = u(xi), Fi = f (xi), U = (U0, U1, . . . , UN) be grid function.

F = (F0, F1, . . . , FN) is the right-hand grid function, D(m)n F is the m-order derivative of the gridfunction F with n-order accuracy. Since (6), we obtain the difference formula
u′(xi) =

Ui+1 − Ui
h

−
h

2!
Fi −

h2

3!
D(1)Fi − ...−

hn−1

n!
D(n−2)Fi +O(hn) (15)

Consider two points x0 = a, xN = b, we obtain
u′(a) =

U1 − U0
h

−
h

2!
F0 −

h2

3!
D(1)F0 − ...−

hn−1

n!
D(n−2)F0 +O(hn) (16)

u′(b) =
UN − UN−1

h
+
h

2!
FN −

h2

3!
D(1)FN + ...+ (−1)n

hn−1

n!
D(n−2)FN +O(hn) (17)

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Eur. J. Math. Anal. 10.28924/ada/ma.4.10 6Substituting formulas (16) and (17) into the boundary condition system, combined with schemenumber 1, we obtain the difference scheme
c0U0 − c1

(
U1−U0
h − h

2!F0 −
h2

3!D
(1)
n−2F0 − ...−

hn−1

n! D
(n−2)
1 F0

)
= C

Ui−1 − 2Ui + Ui+1 = Φi , i = 1, 2, ..., N − 1

d0UN + d1

(
UN−UN−1

h + h
2!FN −

h2

3!D
(1)
n−2FN + ...+ (−1)n h

n−1

n! D
(n−2)
1 FN

)
= D

(18)
The solution of the system is obtained from the pursuit algorithm which has complexity O(N).Since there we have the following theorem:

Theorem 3: The system of difference equations (18) is stable and the solution approximates the
solution of problem (14) with n-order accuracy.The following are some experimental calculation results for the proposed theory.

3. Experimental results and discussions
In this section, we verify the accuracy of the proposed schemes, we let the function ud(x)that satisfies the boundary problem, from there, we determine the right-hand side function f (x),

x ∈ [a, b] and boundary conditions. On grid space ωh, we define the grid function value Ud =

(ud(x0), ud(x1), . . . , ud(xn)) is the exact solution value on the grid ωh. Then, we use the proposedschemes to find approximate solutions U = (U0, U1, . . . , UN), from there, we evaluate the accuracyof the scheme through error ε = ‖U − Ud‖ωh .
3.1. Evaluate the accuracy of scheme 1Example 1: Consider the problem{

u′′(x) = −sinx, x ∈ (0, 1)

u(0) = 0; u(1) = sin(1)Exact solution of the problem is u(x) = sinx .Using scheme 1 combined with the pursuit algorithm, we obtain the grid solution U . The resultsof evaluating the accuracy of scheme 1 are given in table 1
Table 1. Results of evaluating the accuracy of scheme 1 - Example 1

N h8 ‖U − Ud‖ωh h10 ‖U − Ud‖ωh10 1.0000e-008 2.9675e-012 1.0000e-010 1.6467e-01420 3.9063e-011 4.6426e-014 9.7656e-014 1.1343e-01730 1.5242e-012 4.0825e-015 1.6935e-015 3.6987e-01940 1.5259e-013 7.2653e-016 9.5367e-017 2.9933e-02050 2.5600e-014 1.9039e-016 1.0240e-017 4.1543e-02160 5.9537e-015 6.3781e-017 1.6538e-018 8.1987e-02270 1.7347e-015 2.5295e-017 3.5401e-019 2.0679e-022

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Eur. J. Math. Anal. 10.28924/ada/ma.4.10 7

N h8 ‖U − Ud‖ωh h10 ‖U − Ud‖ωh80 5.9605e-016 1.1351e-017 9.3132e-020 6.2629e-02390 2.3231e-016 5.5993e-018 2.8680e-020 2.1680e-023100 1.0000e-016 2.9759e-018 1.0000e-020 8.4722e-024

Figure 1. Error at each point between exact solution and approximated solution - Example 1Example 2: Consider the problem{
u′′(x) = −sinh(x) + 1

9e
−x/3, x ∈ (0, 1)

u(0) = 1; u(1) = sinh(1) + e−1/3

Exact solution of the problem is u(x) = sinh(x) + e−x/3.
Table 2. Results of evaluating the accuracy of scheme 1 - Example 2

N h8 ‖U − Ud‖ωh h10 ‖U − Ud‖ωh10 1.0000e-008 3.3899e-012 1.0000e-010 2.0506e-01420 3.9063e-011 5.3052e-014 9.7656e-014 1.7488e-01730 1.5242e-012 4.6589e-015 1.6935e-015 5.2038e-01940 1.5259e-013 8.3000e-016 9.5367e-017 4.0990e-02050 2.5600e-014 2.1765e-016 1.0240e-017 5.6274e-021

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Eur. J. Math. Anal. 10.28924/ada/ma.4.10 8

N h8 ‖U − Ud‖ωh h10 ‖U − Ud‖ωh60 5.9537e-015 7.2895e-017 1.6538e-018 1.1046e-02170 1.7347e-015 2.8908e-017 3.5401e-019 2.7774e-02280 5.9605e-016 1.2973e-017 9.3132e-020 8.3970e-02390 2.3231e-016 6.3989e-018 2.8680e-020 2.9033e-023100 1.0000e-016 3.4009e-018 1.0000e-020 1.1337e-023
The results in Table 1 and Table 2 confirm that our proposed scheme 1 has determined theapproximated solution of the second-order boundary problem with Dirichlet boundary conditionswith n-order accuracy, where n is the number of neighboring points.
3.2. Evaluate the accuracy of scheme 2Example 3: Consider the problem{

u′′(x) = 6x + 1
8sinh(1 + x/2), x ∈ (0, 1)

u(0)− u′(0) = sinh(1)− 12cosh(1); u(1) + u′(1) = 4 + sinh(3/2) + 1
2cosh(3/2)Exact solution of the problem is u(x) = x3 + sinh(1 + x/2)Using scheme 2 combined with the pursuit algorithm, we obtain the grid solution U . The resultsof evaluating the accuracy of scheme 2 are given in table 3

Table 3. Results of evaluating the accuracy of scheme 2 - Example 3

N h8 ‖U − Ud‖ωh h10 ‖U − Ud‖ωh10 1.0000e-008 2.0382e-013 1.0000e-010 5.3121e-01620 3.9063e-011 3.3597e-015 9.7656e-014 9.1360e-01930 1.5242e-012 2.9998e-016 1.6935e-015 2.5005e-02040 1.5259e-013 5.3835e-017 9.5367e-017 2.0884e-02150 2.5600e-014 1.4183e-017 1.0240e-017 3.1494e-02260 5.9537e-015 4.7654e-018 1.6538e-018 7.0444e-02370 1.7347e-015 1.8943e-018 3.5401e-019 1.6852e-02380 5.9605e-016 8.5167e-019 9.3132e-020 5.3577e-02490 2.3231e-016 4.2068e-019 2.8680e-020 1.8066e-024100 1.0000e-016 2.2381e-019 1.0000e-020 3.0015e-025

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Eur. J. Math. Anal. 10.28924/ada/ma.4.10 9

Figure 2. Error at each point between exact solution and approximated solution - Example 3Example 4: Consider the problem{
u′′(x) = cosh(x) + 1

9e
x/3, x ∈ (0, 1)

u(0)− 12u
′(0) = 1

2 ; 13u(1) + u′(1) = 1
3

(
cosh(1) + e1/3

)
−
(
sinh(1) + 1

3e
1/3
)

Exact solution of the problem is u(x) = cosh(x) + ex/3Using scheme 2 combined with the pursuit algorithm, we obtain the grid solution U . The resultsof evaluating the accuracy of scheme 2 are given in table 4.
Table 4. Results of evaluating the accuracy of scheme 2 - Example 4

N h8 ‖U − Ud‖ωh h10 ‖U − Ud‖ωh10 1.0000e-008 3.7083e-011 1.0000e-010 3.5960e-01320 3.9063e-011 6.1634e-013 9.7656e-014 5.6152e-01630 1.5242e-012 5.5099e-014 1.6935e-015 1.1085e-01740 1.5259e-013 9.8929e-015 9.5367e-017 6.5992e-01950 2.5600e-014 2.6068e-015 1.0240e-017 7.2379e-02060 5.9537e-015 8.7603e-016 1.6538e-018 1.1673e-020

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Eur. J. Math. Anal. 10.28924/ada/ma.4.10 10

N h8 ‖U − Ud‖ωh h10 ‖U − Ud‖ωh70 1.7347e-015 3.4826e-016 3.5401e-019 2.4481e-02180 5.9605e-016 1.5659e-016 9.3132e-020 6.1971e-02290 2.3231e-016 7.7351e-017 2.8680e-020 1.7954e-022100 1.0000e-016 4.1155e-017 1.0000e-020 5.7442e-023
The results in Table 3 and Table 4 also confirm that our proposed Scheme 2 has determined theapproximate solution of the second-order boundary problem with mixed boundary conditions with

n-order accuracy, where n is the number of neighboring points.
4. Conclusion

The main content of article has proposed two difference schemes with high-order accuracy tosolve second-order boundary problems with the Dirichlet boundary condition system and the mixedboundary condition system. The highlight of these two schemes compared to other schemes [3]- [5]is that the two proposed schemes have a simple structure, high accuracy and finding solutions isdone using a pursuit algorithm with the computational complexity is O(N), where N is the numberof grid points. Through the calculation results, it has been confirmed that the two schemes bothprovide approximated solutions with n-level accuracy compared to the grid step with n+1 being thenumber of neighboring points used to determine approximate derivatives of levels. These differenceschemes will allow to improve the accuracy of the solution for the class of nonlinear boundarieswith mixed boundary conditions.
References

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	1. Introduction
	2. Proposed method
	3. Experimental results and discussions
	4. Conclusion
	References

