Efficient Hybrid Block Method For The Numerical Solution Of Second-order Partial Differential Problems via the Method of Lines

This study is therefore aimed at developing classes of efficient numerical integration schemes, for direct solution of second-order Partial Differential Equations (PDEs) with the aid of the method of lines. The power series polynomials were used as basis functions for trial solutions in the derivation of the proposed schemes via collocation and interpolation techniques at some appropriately chosen grid and off-grid points the derived schemes are consistent, zero-stable and convergent. the proposed methods perform better in terms of accuracy than some existing methods in the literature. DOI:10.46481/jnsps.2021.141

The developed differential equations require solutions, either in closed form (analytic form) or in numerical forms. In most cases, closed-form of solutions is rare to come by as there exist limited methods for solving such models in the form of differential equations. This brings to light the use of numerical methods/techniques to solve the modelled problem.
Numerical techniques are numerous and the types know no bounds. They include; The Euler method, Runge-Kutta methods, linear Multistep method, shooting method, Finite difference method, finite element methods, e.g Galerkin method, 26 Spectral element method. Other methods are; Spectral method base on Fourier transformation; Method of lines reduces the PDE to a large system of the ordinary differential equation (ODE) Boundary Element Method (BEM) based on transforming a PDE to an integral equation on the boundary of the domains and it is popular in computational fluid dynamics, the list is endless. Authors who have worked extensively on numerical methods for approximation of solution of a differential equation include but no limited to Brugnano and Trigiante [1], Onumanyi et al. [2,3], Jator [4,5], Fatunla [6], Yusuph and Onumanyi [7], Siraj-ul-Islam, et al. [8], Adewale et al. [9].
We adopt the method of lines approach which is commonly used for solving time-dependent partial differential equations (PDEs), whereby the spatial derivatives are replaced by finite difference approximations see Ngwane and Jator [10].
The Method of Lines (MOLs) allows the conversion of PDEs into ODEs by complete or partial discretisation of the independent variables resulting in algebraic equations. If partial discretization is carried out and with only one remaining independent variable, then this results in the system of ODEs which is an approximation of the original PDE. Thus, one of the underscored features of the MOL is the use of existing, and generally well established, numerical methods for ODEs, for more literature on this approach see Brugnano and Trigiante [1], Ramos and Vigo-Aguiar [11], Cash [12] and Jator and Li [13].

Derivation of the Method
Second order ordinary differential equation of the form is considered subject certain conditions, where a, b are real numbers, f is a continuous function on (a, b) and y ∈ C 2 [a, b]. A 2-step block methods for the problem of the form (1) is considered. The grid points given by x n , x n+1 = x n + h, x n+2 = x n + 2h, are considered for solving the problem in (1) on the interval [x n , x n+2 ]. We assume a trial solution y(x) of (1) by a polynomial p(x) given by which on differentiating yields with the a i ∈ R real unknown parameters to be determined. and m = r + s; r is the number of interpolation points and s is the number of collocation points.

Specification of the method
In this work the interval of integration considered is [x n , x n+2 ], we thus consider two different categories of off-set points viz-aviz the points x i 3 , for i = 1, 2, 4, 5 and x i 4 , for i = 1, 2, 3, 5, 6, 7.

Case 1
Here, we consider the specification where the off-set points are x i 3 , for i = 1, 2, 4, 5. interpolating (2) at the points x i 3 , for i = 1, 2 implies r = 2 and collocating (3) at points x i 3 , for i = 0(1)6 implies s = 7 so that (2) and (3) becomes which on differentiating twice yields From the imposed collocation condition, the following system of algebraic equation is obtained Where y n ≈ y (x n ) , f n ≈ f (x n , y n , y n ). Hence, we state the following theorem without proof.
Theorem 2.1. [14] Let (4) and (5) be satisfied, then the 2-step continuous linear hybrid multistep method is equivalent to the equation where b, A and e are as defined above.

Case 2
Here, we consider the specification where the off-set points are also x i 3 , for i= 1, 2, 4, 5. Interpolating (2) at the points x i+n , for i= 0, 1 implies r= 2 and collocating (3) at points x i 3 , for i = 0(1)6 implies s= 7 so that (2) and (3) becomes which on differentiating yields From the imposed collocation condition, the following system of algebraic equation is obtained Applying the above theorem, the following continuous hybrid method is derived 29 where α and β are function of t given as where t= x−x n−1 h . Evaluating (13) at the points x=x n+ i 3 for i= 1, 2, 4, 5, 6 which is equivalent to t= −2/3, −1/3, 1/3, 2/3, 1 the following main methods are obtained differentiating (14) and evaluating the derivative of (13) at the points x=x n+ i 3 for i= 0(??)6 which is equivalent to t= −1, −2/3, −1/3, 0, 1/3, the following additional methods are obtained Here, it is noteworthy that (9) and (10) are combined to form a block for case 1 while (15) and (16) form another block for case 11. For each case, (1) is solved which is a system of second-order ordinary differential equations resulting from the semidiscretization of a second-order PDE.

Order and Local Truncation Error (LTE)
The LMMs (8), and (13) are said to be of order p if Here C p+µ is the error constant and C p+µ h p+µ y (p+µ) (x n ) is the principal Local Truncation Error (LTE) at the point x n . The C s are given by C 0 =α 0 +α 1 +α 2 +· · ·+α k C 1 = (α 0 +α 1 +α 2 +· · ·+α k ) − (β 0 +β 1 +· · ·+β k ) The LTE associated with any of (8) and (13) is given by the difference operator where y∈C 2 [a, b] is an arbitrary function. Expanding (17) in Taylor's series about the point x n , the expression is obtained as: Expanding each scheme in (9) and (10), the following principal truncation errors are obtained: The above blocked method (9) and (10) is of uniform order p= 7 Expanding each scheme in (15) and (16), the following principal truncation errors are obtained: The above blocked method (15) and (16) is of uniform order p= 7 The LMM (8) (same for (13)) is said to be consistent if it has order p≥1 and the first and second characteristic polynomials which are defined respectively, as and where r is the principal root, satisfy the following conditions: Henrici [15], Lambert [16]. Consider the main method in (9) The condition (21) is satisfied. the first characteristic equation for (9) is given as: The condition (21) is satisfied. the first characteristic equation for (15) is given as: Here ρ(1) = 0, ρ (1) = 0. Therefore, (22) is satisfied. The second characteristic polynomial for (15)  For block method (15)-(16) similar operation is carried out; A numerical method is zero-stable if the solutions remain bounded as h→0, which means that the method does not provide solutions that grow unbounded as the number of steps increases, Modebei et al. [17]. To show the zero-stability of the block method (36), we take h→0 the method may be rewritten in matrix form as For method (36) A 0 =I 12×12 identity matrix and A 1 =I 12×12 matrix given by The characteristic polynomial of the matrix A 11 is given as |A 11 −λI|, that is λ 5 (λ−1)= 0 with root λ j = 0 for j= 1, . . . , 5 and λ 6 = 1.

Convergence of the Methods
Definition 3.2. Convergence: An LMM is said to be convergent if and only if it is consistent and zero-stable.
By the above definition, the derived hybrid methods are convergent.

Numerical Examples
In this section, the performance of the developed two-step hybrid block scheme is examined. the exact and approximate solution are tabulated. The tables below show the numerical results of the newly developed scheme with the exact solution for solving the problem and the result of the developed scheme are more accurate than existing methods. For simplicity, method in (9)-(10) would be termed Hybrid 2-step Block Method 1 (H2BM1), and method in (15)-(16) would be termed Hybrid 2-step Block Method 2 (H2BM2) Example 1. Consider the PDE (Ngwane and Jator [10]). κ ∂y 2 ∂x 2 − ∂y ∂t = 0 y(0,t) =y(1,t) = 0, y(x, 0) = sinπx+sinωπx, κ>1 The analytic solution is y(x, t) =e −π 2 κt sinπx+e −ω 2 π 2 κt sinπx Following Ngwane and Jator [10], (38) becomes   BHSDA is L -Stable Block Hybrid Second Derivative Algorithm in Ngwane and Jator [10]. Tables 1 and 2 shows the comparison of exact solution and the mothers H2BM1 and H2BM2 respectively. For example 1. Table 3 shows the comparison of maximum errors obtained for example 1 using the derived methods and the method in Ngwane and Jator [10]. This shows the superiorly of the derived methods over existing methods. Figure 1 show the surface plots for the exact solution and Numerical solutions for example 1. Example 2. Consider the PDE ( [14]):    The analytic solution is y(x, t) = sin4πxsin4πt. Following Ngwane and Jator [10], (38) becomes BVM and BUM are Boundary Value Methods and the Block Unification Methods in Biala [14].
Tables 4-5 shows the comparison of the exact solution and the methods H2BM1 and H2BM2 respectively for example 2. Table 6-7 shows the maximum error and CPU time obtained for different methods. Table 9 shows the maximum error and CPU time obtained in Biala [14].  This show that the derived methods performs accurately, superiorly and affluently in terms of the computer time, and errors obtained in examples 2. Figure 2 shows the surface plots for the exact and Numerical solution for examples 2. Example 3. We consider the PDE (Jator [15]).
SBVM is symmetric boundary value method in Jator [15]. Table 8 shows the exact and numerical solution using the different methods for example 3 while Table 9 shows error obtained for example 3. Figure 3 shows the surface plots for the Exact and numerical solutions of H2BM1 and H2BM2 for Example 3

Conclusion
The development of some numerical schemes has been proposed in this work. This was developed via the interpolation and collocation techniques using power series function as trial solutions. The methods were effectively illustrated some partial differential equations (PDE) and the results obtained were accurate. The analysis of the new methods showed that all satisfy the properties of numerical methods for the solution of differential equations. Namely, Consistency, Zero-Stability, Continuity and convergence.