ImplicitTwo-PointBlockMethod forSolvingFourth-Order Initial Value Problem Directly with Application

)is paper proposes an implicit block method with two-point to directly solve the fourth-order Initial Value Problems (IVPs).)e implicit block method is derived by adopting Hermite interpolating polynomial as the basis function, incorporating the first derivative of f(t, y, y′, y′′, y′′′) to enhance the solution’s accuracy. A block formulation is presented to acquire the numerical approximation at two points simultaneously. )e introduced method’s basic properties, including order, zero stability, and convergence, are presented. Numerical experiments are carried out to verify the accuracy and efficiency of the proposed method compared with those of the several existing methods. Application in ship dynamics is also presented which yield impressive results for the proposed two-point block method.


Introduction
Higher order differential equations (ODEs) can be used to model problems arising from the field of applied sciences and engineering [1]. e problem of static deflection of a uniform beam which can be modelled as a fourth-order initial value problem (IVP) is a good example of a real problem in engineering [2,3]. e classical way of solving them is by reducing the equation into the system of firstorder ODEs, but this process is too rigorous compared with the direct methods [4,5]. In addition, many researchers have presented direct methods to avoid the reduction effort [6][7][8][9][10][11][12][13][14][15]. To enhance the efficacy of numerical methods, the block method is introduced with the idea of producing simultaneously r-point of the approximate solutions at one time step. Kuboye and Omar [7] and Jacob [8] introduced efficient zero-stable numerical block methodologies for fourth-order initial value problems (IVPs). For block multistep method, approximations are used to generate the approximate solutions at the new block. Omar and Kuboye [9] proposed a block method for directly solving general fourth-order IVPs by increasing the step number; Adeyeye and Omar [14] also presented a direct six-step block method. Recently, researchers have been exploring numerical methods with more functional evaluations with the aim of obtaining numerical solutions with very high precision. is is how hybrid methods in nature arises. Jator [16] and Yap and Ismail [10] developed four-step hybrid block methods, and Abdelrahim and Omar [11] generalized three off-step points. Kayode et al. [12] modified the implicit hybrid block method to directly solve the IVPs associated with fourthorder ODEs. Generally, collocation and interpolation techniques are utilized as direct methods. e points need to be collocated and interpolated after which a system of linear equations must be resolved in order to obtain the method's coefficients. erefore, we developed a two-point block the implicit method by using a strategy in our proposed method which can be implemented in a straightforward manner. e method replaces the function by interpolating and integrating the polynomial. Our derivation only requires us to do interpolation and integration to obtain the coefficients. e method also involves the fifth derivative of the solution to equation (1), which aims at acquiring better accuracy. In general, the accuracy of a method increases with increase in the order of the method. But, with the idea of incorporation of higher derivative of the solution in the process, higher and better accuracy is achieved without corresponding increase in the order of the method. We are concerned with the development of the numerical solution of IVPs for fourthorder ODEs of the form: where the fifth derivative of the solution to equation (1) is We assume that f in equation (1) is differentiable to a desired order in region R and f(t, y, y ′ , y ′′ , y ′′′ ) satisfies Lipchitz condition in its second, third, fourth, and fifth terms as follows: for all points (t, y i , y ′ , y ″ , y ‴ ), (t, y, y i ′ , y ″ , y ‴ ), (t, y, y ′ , y i ″ , y ‴ ), and (t, y, y ′ , y ″ , y ‴ i ), i � 1, 2 in the region R. en, the IVPs in equation (1) have a unique solution in R ( [17,18]). e paper is structured in the following manner: in Section 2, a discussion is undertaken about the derivation of the block method of order six (I2PBDO6) along with the basic idea on how the block works. e implementation of this method to solve general fourth-order ODEs is proposed in Section 3. Numerical experiments and the application of the ship dynamics problem are given in Section 4 to show the accuracy and efficiency of the proposed method. e study's conclusion is finally provided in Section 5.

Derivation of Method
is section depicts the proposed method's derivation based on Hermite Interpolating Polynomial P 2 (t), which interpolates f(t, y, y ′ , y ″ , y ″′ ) at two points. e polynomial has the form where and h � (b − a/n); L (i,k) (t) is the generalized Lagrange polynomial, i � 0, 1, . . . , n, k � 0, 1, . . . , m i , and n is a positive integer. e method computes the approximate solutions y n+1 and y n+2 concurrently at t n+1 and t n+2 , where t n becomes the starting point and t n+2 is the last point in the block with step size 2h. e evaluation solution of y n+2 at the point t n+2 will be used as the next iteration's initial value: e approximate solution y n+1 at the point t n+1 can be retrieved by integrating equation (1) once, twice, thrice, and four times, in relation to t over the interval [t n , t n+1 ]. e following formula can be reached: en, f(t, y, y ′ , y ″ , y ″′ ) in equations (9)-(12) will be replaced by Hermite interpolating polynomial P 2 (t).
Taking t � t n+2 + sh leads to s � (t − t n+2 /h) and dt � hds and then making changes in the integration limit from −2 to −1, we have Mathematical Problems in Engineering where L 0,0 � 1 4 (s + 1) 2 s 2 (1 + 3(2 + s)), Evaluating the integrals in equations (13)- (16) gives Now, we have four formulas, one for the approximate solution and the others for the approximation of the first, second, and third derivatives of the proposed solution at the block's end point [t n , t n+1 ]. erefore, we need to consider the evaluation of p 2 (t) at the point y n+2 over [t n+1 , t n+2 ] to have a two-point implicit block method. By applying the same technique as for the formula for y n+1 , we have the following formula at t n+2 : where β, c, and δ are defined as It is possible to define the linear operator that is linked to equation (26) as Expanding equation (28) in the Taylor series yields e linear operator and the new method have order p if C 0 � C 1 � · · · � C p � · · · � C p+3 � 0, C p+4 ≠ 0, which is the error constant. Hence, in our method, C 0 � C 1 � · · · � C 9 � 0 and C 10 � [1.058199 × 10 − 4 , 5.78707× 10 − 5 , 1.763674× 10 − 5 , 3.858 × 10 − 6 , 1.0581959× 10 − 4 , 4.795011 × 10 − 5 , 1.267633 × 10 − 5 , 2.479443 × 10 − 6 ] T . us, it can be concluded that the order of the proposed method is 6.

Zero Stability.
e method (18)-(25) is deemed zerostable in case the roots z i � 1, 2, . . . , N of the first characteristic polynomial ρ(z) � det(zA (0) − A (1) ) is found to satisfy |z i | ≤ 1. Now, in our proposed method, we will use the following technique to find the matrix of the first characteristic polynomial.

Results and Discussion
Within this section, we tested the I2PBDO6 method to solve single as well as a system of initial value challenges in the form of equation (1). We will see how the method of twopoint block (order 6) produces comparable results to the existing direct methods of similar characteristics. Most of the existing methods are of higher order in comparison to the proposed method. e following notations will be used in the tables: (i) I2PBDO6: the direct implicit two-point block method proposed in this paper, order 6 (ii) MCM: direct maximal order multiderivative collocation method [ Exact solution: y ″′ (0) � 1. (40)

Problem 7
Mathematical Problems in Engineering y (4) 1 � e 3t y 4 , y 1 (0) � 1, Exact solution: e aim of this study was to demonstrate the accuracy, efficiency, and applicability of the proposed I2PBDO6 method. Tables 1-4 show the absolute errors at different points as well as different step size. It is remarkable that the errors are zero at some points of the interval which means that the computed solutions are identical to the exact solutions with 20 digits at least. e two-point block method can be seen to significantly outperform MCM [6], ZSB7 [7], LIBO7 [14], ZSDM6 [8], and IHB8 [12] with regard to accuracy. For Problems 3 to 5, the results of the proposed I2PBDO6 method are tabulated alongside others in Tables 5-7. We have considered the absolute errors along the interval using different number of steps in terms of accuracy as well as steps taken. It can be observed that increasing the number of steps leads to decrease in the error except that in Table 6 at the number of Step 5004 and 50004, since Problem 4 involves exponential function which requires an infinite number of digits for numerical representation. However, these numbers must be represented by a finite number of digits in computer for numerical computation that causes round-off errors. us, at a large number of steps such as 5004 and 50004, a sequence of many operations and each subject to a small rounding error is obtained, where the error accumulates and affects the accuracy of the final solutions. e proposed method shows very good performance compared with FDM6 [16], BHC8 [10], FSHB8 [11], SSB8 [9], and LIBO7 [14]. e system in Problem 6 and Problem 7 has integrated in the interval t ∈ [0, 10] and t ∈ [0, 3], respectively. A comparison of numerical outcomes is made with the solution of the same group of systems when they are solved by using RKTF5 [13] and RKF5 [19] methods. We have solved these systems to consider the maximum errors and the number of function evaluations at different step   Overall, the proposed method I2PBDO6 of order 6 has shown remarkable convergence since the approximate solution is very close to the exact solution. Moreover, the method is more efficient compared with the all present methods whose order is nearly equal to or greater as compared with that of the proposed method. e data given in Tables 1-9 and Figures 1 and 2 show the superiority of I2PBDO6 with regard to efficiency, accuracy, and total number of steps taken at different points of t or different step sizes.

Application to Ship Dynamics Problem.
We take into consideration a physical challenge from the ship dynamics as shown in Wu et al. [20], where a sinusoidal wave passes via a ship or offshore structure, with the accompanying fluid actions varying with time t. In the aforementioned study [20], the differential equation is defined as with y(0) � 1, where ε � 0 with regard to the theoretical solution, On the other hand, the      theoretical solution can be seen to be indeterminate when ε ≠ 0 as can be seen in [21]. Numerical investigations to resolve the ordinary differential equations of the fourthorder have been expanded for resolving a problem from ship dynamics. Numerical consideration was offered when ε � 0 and ε � 1 with Ω � 0.25( � 2 √ − 1) as can be seen in Twizell [21] and Cortell [22]. However, Twizell [21] and Cortell [22] derived the ODE method by lowering the equations to system of first-order (ODEs) rather than directly resolving it. Moreover, Cortell [22] introduced the extension of the classical RK method. Figure 3 depicts the exact and approximate solutions by using I2PBDO6 for the application of ship dynamics problem with ε � 0. However, the results of equation (52) From Figures 3 and 4, it is clear that the solutions that are obtained by I2PBDO6 strongly agree with not only the exact solutions but also with those of Mathematica inbuilt package NDSolve, respectively.

Conclusions
In this article, we proposed a two-point block method with extra derivative to directly solve initial value problems of the fourth order. e derivation of the new method can be easily implemented. is implicit block method is found to satisfy the property of convergence as suggested by the significant enhancement with regard to accuracy in the numerical results. erefore, the proposed method I2PBDO6 is suitable for attaining high precision when solving fourth-order ODEs.

Data Availability
e data used to support the findings of this study are included within the article.