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Numerical methods for initial value problems in ordinary differential equations
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... Taylor's series was applied, as the starting values in the implementation of the scheme. The major disadvantage of this method stated in literature has been the use of predictors of lower order to implement the scheme reported in [16][17][18][19][20][21][22][23]. Furthermore, in Ref [20], the authors developed a zerostability method that directly solves fourth-order ordinary differential equations. ...
... The major disadvantage of this method stated in literature has been the use of predictors of lower order to implement the scheme reported in [16][17][18][19][20][21][22][23]. Furthermore, in Ref [20], the authors developed a zerostability method that directly solves fourth-order ordinary differential equations. The methods used were collocation of differential systems and interpolation of approximate solutions to problems using power series as basis functions. ...
... where ( ) is any continuously differentiable test function on the interval [ , ]. Expanding ( + ℎ) and ′′ ( + ℎ), ′′′ ( + ℎ), = 0,1, … , in (75) Taylor series about and collecting like terms in ℎ and gives; [ ( ); ℎ] = 0 ( ) + 1 ℎ ′ ( ) + ⋯ ℎ ( ) + +1 ℎ +1 +1 ( ) + +2 ℎ +2 +2 ( ) + ⋯ Definition 1: [20] The term +4 is called an error constant, meaning that the local truncation error is given as ...
This article focuses on the development, theoretical analysis, and implementation of a four-level predictor-corrector method with an improved strategy for solving fourth-order ordinary differential equations using initial conditions. When developing the method, Chebyshev polynomials of the first kind were adopted as the basis functions for solving the IVP. Chebyshev polynomials of the first kind were interpolated at some selected grid and off-grid points, and the fourth derivative of the approximate solution was collocated at all grid and off-grid points. The methods was derived and implemented in such a way that the correctors are of the same order with the predictors in other to overcome setbacks associated with existing predictor-corrector, thus enhanced a better accuracy and stability. Theoretical analysis of the methods were validated to ensure that the methods are usable and efficient. The methods was applied to three numerical test problems in order to establish the accuracy of the methods. The results show better accuracy over some existing approach in the literature.
... The primitive approach of reducing higher order ODEs to first-order before solving them numerically has been extensively discussed by several authors such as [14] and [6] with resulting setbacks such as wastage of time and computational burdens. To cater for these setbacks, several numerical methods have been developed (see [1][2], [10][11][12], [16][17][18][19], [20,21], [22][23][24][25][26][27][28][29][30][31][32][33][34][35]). ...
... Following [10] and [17], we define the local truncation error associated with equation where y(x) is an arbitrary function, continuously differentiable on [ a, b ]. Expanding (3.2) in Taylor series about the point x, we obtain the expression ...
... and solving for R , the values of R are obtained as 0, 0 and 1. According to [10] and [11], the block formula represented by equation (3.3) is zero-stable, since from equation (3.4), ρ(R) = 0 , satisfy |R j | ≤ 1, j = 1 and for those roots with |R j | = 1 , the multiplicity does not exceed two. ...
... The implicit schemes from equation (6) Following Refs. [2,11], we define the Local Truncation Error(LTE) associated with equation (22) by difference operator; ...
... Theorem 1. Zero-stability [1,11]: A block method is said to be zero stable if as h → 0, the roots r j , j = 1(1)k of the first characteristic polynomials ρ(r) = 0 that is ρ(r) = det A (0) R k−1 = 0, satisfying |R| ≤ 1, must be simple. ...
... When these conditions are applied to the main scheme, it is found to be consistent. [11]: The necessary and sufficient condition for a linear multistep method to be convergent is for it to be consistent and zero stable. Thus our block method is convergent since it is zero stable and consistent. ...
The numerical solution of second-order ordinary differential equations (ODEs) is examined in this work through a four-step linear multistep method. It employs a combination of trigonometric and polynomial functions as the approximate solution to the general second-order ordinary differential equations (ODEs). The method was developed using interpolation and collocation techniques. This methodology involves interpolating the basis function at two points and subsequently collocating it across all points, ensuring a robust scheme. To assess its efficacy, we solved three initial value problems (IVPs) associated with stiff differential equations. Through this examination, we established the method’s core characteristics: consistency, zero stability, and consequently convergence. This thorough analysis demonstrates its reliability and suitability for resolving second-order ordinary differential equations. The comparison of our newly derived block method against existing approaches reveals its superiority. Our method’s performance, evaluated across a spectrum of stiff second-order ordinary differential equations, surpasses the outcomes obtained from established authors. This substantiates its efficiency and effectiveness in addressing these mathematical challenges. This study marks a significant advancement by introducing a robust approach that not only accurately solves second-order ordinary differential equations but also streamlines the computational process. By integrating trigonometric and polynomial functions and leveraging interpolation and collocation techniques, our method stands out for its accuracy, stability, and convergence properties, offering a promising avenue for future research in this domain.
... Some researchers have treated higher order ordinary differential equation (1) directly without going through the process of reduction because it is difficult, among others are [1][2][3][4][8][9][10][11]. We notice that direct solutions is competent than reduction method in term of accuracy. ...
... (ξ) 7 7 ! (ξ) 8 8 ! (ξ) 9 9 ! ...
... The concept of A-stability according to [8] is discussed by applying the test equation. ...
The numerical application of higher order linear block method for the direct solution of fourth order initial value problems was proposed using the linear block algorithm, where the methods applied in block form. The method is zero-stabile, consistent and convergent when analyzing the properties of the method. The mathematical example solved using the method is effective, suitable, and acceptable for solving fourth order initial value problems. The method is also compared with existing work when solving similar systems of differential equation and obviously, the method performs better than those in literature and textual shown.
... The added pressu applied at each side of the openings can be calculated iteratively at each time step usin nonlinear system of equations [25,37]. For the case in Figure 3, the added pressure in co partment B ( corresponds to an additional virtual added column of water , the s tem of equations reduces to a single equation, and the added pressure is calculated follows: In terms of the implementation of the time domain procedure, different time-mar ing schemes can be used, e.g., implicit Euler [38] or the more common Runge-Kutta 4 order explicit scheme [39]. An adaptive time step can also be used [40]. ...
... The added press applied at each side of the openings can be calculated iteratively at each time step us nonlinear system of equations [25,37]. For the case in Figure 3, the added pressure in partment B ( corresponds to an additional virtual added column of water , the tem of equations reduces to a single equation, and the added pressure is calculate follows: In terms of the implementation of the time domain procedure, different time-ma ing schemes can be used, e.g., implicit Euler [38] or the more common Runge-Kutta order explicit scheme [39]. An adaptive time step can also be used [40]. ...
... It is very unlikely that it happens precisely at the end of a time step. As suc avoid spurious overfilling of a tank, a possible procedure is to monitor the filling stat each compartment and, when overfilling is detected, perform a series of regress-prog In terms of the implementation of the time domain procedure, different time-marching schemes can be used, e.g., implicit Euler [38] or the more common Runge-Kutta 4th-order explicit scheme [39]. An adaptive time step can also be used [40]. ...
The timely and precise prediction of flooding progression and its eventual outcome in ships with breached hulls can lead to dramatic improvements in maritime safety through improved guidance for both emergency response and ship design. The traditional approach to assessing damage-induced flooding in both these stages, which also fully complies with statutory rules, is through static calculations. On the other hand, the application of models that simulate the flooding progression and the behaviour of flooded ships from, or close to, first principles allows for increased accuracy of the modelling of the phenomenon. This increase in accuracy can then be used to support advanced design for safety procedures. Furthermore, it can considerably enhance a ship’s capability for damage identification and inference-based logic for emergency decision support systems and marine accident response in general. This paper conducts a review of selected state-of-the-art methods, procedures, and case studies in recent years which aimed to model progressive flooding and damage ship behaviour and provide some explanations of fundamentals. Applications related to damage identification, the prediction of outcome/situation awareness, and flooding emergency response are also briefly discussed. The paper concludes with a brief reflection on salient gaps in the context of accelerating the development of these methods and their applicability.
... This led us to search and developped some one-step methods which we believed can provide solution to singular problems. Before designing our formulae, we considered many methods and we were motivated by the striking proposal made by Evans and Sangui (1986), Aashikpelokhai (1991), Fatunla (1980Fatunla ( , 1988 to study Runge-Kutta method of order 4. *Corresponding author. E-mail: isaacaigbedion@yahoo.com. ...
... In real life, problems with singularities abound in physical phenomena such as simulation, control theory, economics analysis and production processes, oil spillage, chemical kinetics, electrical network, tunnel switching, petroleum exploration, population problems, the states of the national economy nuclear reactor control as indicated in the work of Ascher and Mattheij (1988), Edsberg (1988), Fatunla (1980Fatunla ( , 1987aFatunla ( and 1988, Kaps (1984), Lee and Preiser (1978), Norelli (1985), Parker (1982) and Robertso (1976) are problem whose differenttial equations are stiff, singular or oscillatory. ...
... According to Fatunla (1988) the Conventional numerical integrators are, in general, formulated on the basis of polynomial interpolation, with the tacit assumption that the IPV in (1) satisfies the hypothesis of the existence and uniqueness theorem. ...
A new method for solving singular initial value problems in ordinary differential equations is developed and implemented using the test problem in (1). Results generated through a FORTRAN program were found to be highly accurate and consistent with minima errors in the solution of some selected singular ivps. A comparison of the results generated from the formula was carried out with other Runge Kutta formulae and were found to compare favourably well.
... From formula (2), the numbers b 1 , b 2 , . . . , b s and c 1 , c 2 , . . . ...
... If a ij = 0, j ≥ i, i = 1(1)s, then Y i is said to be defined explicitly so that formuale (2) and (3) form an explicit Runge-Kutta method. In most cases, explicit Runge-Kutta method is preferable because it allows explicit stage-by-stage implementation which is very easy to program using computer. ...
... However, numerical analysts also aware that the computational costs involving function evaluations increases rapidly as higher order requirements are imposed, [1]. Another disadvantage of explicit Runge-Kutta method is that it has relatively small interval of absolute stability renders them unsuitable for stiff initial value problems, [2]. In view of this, we are thus taking interest in implicit Runge-Kutta methods. ...
In this paper, four new implicit Runge-Kutta methods which based on 7-point Gauss-Kronrod-Lobatto quadrature formula were developed. The resulting implicit methods were 7-stage tenth order Gauss-Kronrod-Lobatto III (GKLM(7,10)-III), 7-stage tenth order Gauss-Kronrod-Lobatto IIIA (GKLM(7,10)-IIIA), 7-stage tenth order Gauss-Kronrod-Lobatto IIIB (GKLM(7,10)-IIIB) and 7-stage tenth order Gauss-Kronrod-Lobatto IIIC (GKLM(7,10)-IIIC). Each of these methods required 7 function of evaluations at each integration step and gave accuracy of order 10. Theoretical analyses showed that the stage order for GKLM(7,10)-III, GKLM(7,10)-IIIA, GKLM(7,10)-IIIB and GKLM(7,10)-IIIC are 6, 7, 3 and 4, respectively. GKLM(7,10)-IIIC possessed the strongest stability condition i.e. L-stability, followed by GKLM(7,10)-IIIA and GKLM(7,10)-IIIB which both possessed A-stability, and lastly GKLM(7,10)-III having finite region of absolute stability. Numerical experiments compared the accuracy of these four implicit methods and the classical 5-stage tenth order Gauss-Legendre method in solving some test problems. Numerical results revealed that, GKLM(7,10)-IIIA was the most accurate method in solving a scalar stiff problem. All the proposed methods were found to have comparable accuracy and more accurate than the 5-stage tenth order Gauss-Legendre method in solving a two-dimensional stiff problem.
... where f is continuous in [ ] , a b , such equation in (1) is often encountered in areas such as control theory, chemical kinetics, circuit theory, biological sciences and many others. The fact that most often, this class of equations cannot be solved analytically, then the approach in developing several numerical methods to approximate the solution of problem (1) comes in [9,12]. Some approaches to this alternative method (i.e. ...
... Hence, equations (6) to (9) are the required one-step hybrid block method for the solution of equation (1). Similarly, by expanding equation (3) and after some algebraic simplifications, we obtained the two-step hybrid block method as: ...
... Thus, the corrector of the one-step hybrid block method is normalized to give the first characteristic polynomial [10]. Putting equation (6) to (9) in matrix form as a block we obtain: The following matrix difference equation will be in the form: ...
In this paper, we developed a numerical Algorithm for one and two-step hybrid block methods for the numerical solution of first order initial value problems in ordinary differential equations using method of collocation and interpolation of Taylor's series as approximate solution to give a system of non linear equations which was solved to give a hybrid block method. To further justify the usability and effectiveness of this new hybrid block method, the basic properties of the hybrid block scheme was investigated and found to be zero-stable, consistent and convergent. The derived scheme was tested on some numerical examples and was found to give better approximation than the existing methods. The errors displayed after solving some selected initial value problems, revealed that, it is better to increase L (Derivative) rather than the step length k as shown in our numerical results. Also, It was difficult to satisfy the zero-stability for larger k. In addition, the new method converges faster with lesser time of computation, which address the setback associated with other methods in the literature. Finally, the new method has order of accuracy for one-step as order Ten while order Eighteen for two-step.
... The advantages of LMMs over single step methods have been extensively discussed in [12]. Block integrators for solving ODEs have been developed by W. E. Milne [13] who employed them as starting values for predictor-corrector algorithms, [14] improved Milne's method in the form of implicit integrators and [15] also contributed greatly to the development and application of block integrators. ...
... The new numerical integration scheme obtained was particularly suited to solve oscillatory and exponential problems. This method was in tune with those developed by [15,17], and [19]. In this work, we derive an order seven block method to solve the problems in [20] which were solved by an interpolating polynomial. ...
... Note that in the tables following, OTM is an abbreviation for the One-Tenth Step Method and Error= |Exact Solution -Computed Result| (7) to (14) gives the result summarized in Table 1. (7) to (15) gives the result summarized in Table 2. (7) to (16) gives the result summarized in Table 1. (7) to (17) gives the result summarized in Table 1. ...
... Using the approach described in [6] and [13]. In equation ( Definition 4.2. ...
... In equation ( Definition 4.2. According to [6], if c 0 = c 1 = c 2 = c 3 = ⋯ = c p = 0, c p+1 ≠ 0, then the linear difference operator and the corresponding continuous LMM are considered to be of the order p. The c p+1 is termed the error constant and the local truncation error is defined by Region of Absolute Stability of the Method Definition 4.2 [14]: A region of absolute stability is one in which r = λh in the complex z plane. ...
... This gives us the absolute stability region shown in Figure 4.1 below. According to Figure 4.1, the new hybrid block method is effective in handling stiff problems since its RAS (Region of the Absolute Stability) is unbounded [6]. A numerical scheme is considered A-stable if its region of absolute stability covers the entire complex plane ℂ, which is defined as, . . ...
This paper presents the derivation and implementation of a hybrid block method for solving stiff and oscillatory first-order initial value problems of ordinary differential equations (ODEs). The hybrid block method was derived by continuous collocation and interpolation using combined Hermite polynomials and exponential functions as the basis function to produce a continuous implicit Linear Multistep Method (LMM) of order nine and implement in block form. The basic properties of the derived method were studied, and the hybrid block integrator was demonstrated to be zero-stable, convergent, consistent, and to have an A-stable region of absolute stability, which made it suitable for stiff and oscillatory ordinary differential equations. The use of a combined basis in the generation of LMMs is worthy of universal acceptance. The technique indicates that, utilizing an interpolation and collocation approach, continuous LMMs can be derived from combinations of any polynomials and exponential functions. On two sampled stiff and oscillatory problems, the new integrator was tested. The numerical results indicate that our new hybrid block integrator is computationally efficient and outperforms existing methods in terms of accuracy
... An alternative way to study the evolution operator is direct numerical integration. There is a vast number of different numerical algorithms [9,10], which may be used for an investigation of the evolution operator. In many applications, the direct numerical integration using high-performance algorithms is preferable than the perturbation approach. ...
... Such a choice of the projection operators allows to drop off terms proportional to QI in Eqs. (10) and (11), because of they are equal to zero. ...
The not necessarily unitary evolution operator of a finite dimensional quantum system is studied with the help of a projection operators technique. Applying this approach to the Schr\"odinger equation allows the derivation of an alternative expression for the evolution operator, which differs from the traditional chronological exponent. An appropriate choice of projection operators results in the possibility of studying the diagonal and non-diagonal elements of the evolution operator separately. The suggested expression implies a particular form of perturbation expansion, which leads to a new formula for the short time dynamics. The new kind of perturbation expansion can be used to improve the accuracy of the usual chronological exponent significantly. The evolution operator for any arbitrary time can be efficiently recovered using the semigroup properties. The method is illustrated by two examples, namely the dynamics of a three-level system in two nonresonant laser fields and the calculation of the partition function of a finite XY-spin chain.
... Here, we remark that the mentioned numerical method is a powerful tool and has been used very effectively in the literature. The convergence and stability analyses of the said method, for classical as well as fractal fractional derivatives, were given in [50,51] recently. Since the model (1) seems inconsistent, we therefore modified and investigated it under the fractal fractional derivative of order σ, ξ ∈ (0, 1] by using Λ = nβ n . ...
... Here, we state the the convergence and stability of the used numerical scheme for the ordinary initial value problems that have been given in [50]. But, in the case of fractal fractional ABC derivatives, the detailed analysis about convergence and stability have been given by Khan and Atangana [51] recently. ...
In the last few years, the conjunctivitis adenovirus disease has been investigated by using the concept of mathematical models. Hence, researchers have presented some mathematical models of the mentioned disease by using classical and fractional order derivatives. A complementary method involves analyzing the system of fractal fractional order equations by considering the set of symmetries of its solutions. By characterizing structures that relate to the fundamental dynamics of biological systems, symmetries offer a potent notion for the creation of mechanistic models. This study investigates a novel mathematical model for conjunctivitis adenovirus disease. Conjunctivitis is an infection in the eye that is caused by adenovirus, also known as pink eye disease. Adenovirus is a common virus that affects the eye’s mucosa. Infectious conjunctivitis is most common eye disease on the planet, impacting individuals across all age groups and demographics. We have formulated a model to investigate the transmission of the aforesaid disease and the impact of vaccination on its dynamics. Also, using mathematical analysis, the percentage of a population which needs vaccination to prevent the spreading of the mentioned disease can be investigated. Fractal fractional derivatives have been widely used in the last few years to study different infectious disease models. Hence, being inspired by the importance of fractal fractional theory to investigate the mentioned human eye-related disease, we derived some adequate results for the above model, including equilibrium points, reproductive number, and sensitivity analysis. Furthermore, by utilizing fixed point theory and numerical techniques, adequate requirements were established for the existence theory, Ulam–Hyers stability, and approximate solutions. We used nonlinear functional analysis and fixed point theory for the qualitative theory. We have graphically simulated the outcomes for several fractal fractional order levels using the numerical method.
... "The previous efforts have been made by eminent researchers to solve higher order initial value problems specifically, the second order ordinary differential equation. In exercise, this class of problem (1.1) is usually reduced to system of first order differential equation and numerical methods for first order ODEs then employ to solve them, these researchers" [1][2][3] showed that "reduction of higher order equations to its first order has a serious implication in the results; hence it is necessary to modify existing algorithms to handle directly this class of problem (1.1)". ...
... To determine the region of absolute stability of the block method, the methods that compare neither the computation of roots of a polynomial nor solving of simultaneous inequalities was adopted. Thus, the method according to [1] is called the boundary locus method. Applying this method we obtain the stability polynomial as ...
The manuscript proposed a physical application of single step block method using the interpolation and collocation procedure for the direct solution second order physical oscillatory initial value problem. The properties of the new method which include error constant, order, zero-stability, consistency and convergent are established and satisfied. The new method was tested on some second order oscillatory initial value problems and compared with the existing works in literature, and later the new method revealed its superiority by producing less error if compared. Therefore, the new method does not required much computation when compared with predictor corrector methods.
... The hybrid block method [16] is said to be consistent if it has an order more than or equal to one. Therefore, our method is consistent. ...
... Hence our method is zero-stable [16,17]. ...
In this paper, we develop the double step hybrid linear multistep method for solving second order initial value problems via interpolation and collocation method of power series approximate solution to give a system of nonlinear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic numerical properties of the hybrid block method was established and found to be zero-stable, consistent and convergent. The efficiency of the new method was conformed on some initial value problems and found to give better approximation than the existing methods.
... Many algorithms have been proposed in scholarly works, taking into account the specific characteristics and form of the differential equations that need to be solved. Numerous examples of these algorithms can be found in the literature, such as [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], among others. Fadugba and Idowu [27] applied a new numerical method with third-order accuracy to solve initial value problems (IVPs) in ODEs, analyzing its properties in the process. ...
... This shows that the derived method is absolutely stable. Alternatively, the stability property of the derived method is given by the following result [18]. ...
A. This work introduces and analyzes a novel higher order explicit method for solving physical models in real-life situations. This new method is derived via a hybrid interpolating function which is the combination of Chebyshev polynomials of first kind and exponential function. Its properties, including consistency, local truncation error, stability, order of accuracy, and convergence, are thoroughly examined and studied. To evaluate its effectiveness, the proposed method is applied to four numerical examples derived from real-world scenarios. Furthermore, this study compares the results obtained from the new numerical method with those of the well-known PJS method [28], in terms of the exact solution. The study concludes that the method provides accurate solutions and can be considered as one of the suitable approaches for solving first-order initial value problems (IVPs). 2020 Mathematics Subject Classification. 34A12, 65L05, 65L20, 65L70.
... Techniques like Taylor's series, numerical integration, and collocation methods, which are constricted by a supposed order of convergence, can be used to generate Linear Multistep Methods (LMMs) for solving ODEs (Ajileye et al., 2018). W. E. Milne devised block integrators for solving ODEs (Anake et al., 2012), employed them as the basis for predictor-corrector algorithms (Dahlquist, 1956), significantly improved Milne's technique utilizing implicit integrators (Fatunla, 1988), and made crucial contributions to the development and application of block integrators (Fatunla, 1988). ...
... Techniques like Taylor's series, numerical integration, and collocation methods, which are constricted by a supposed order of convergence, can be used to generate Linear Multistep Methods (LMMs) for solving ODEs (Ajileye et al., 2018). W. E. Milne devised block integrators for solving ODEs (Anake et al., 2012), employed them as the basis for predictor-corrector algorithms (Dahlquist, 1956), significantly improved Milne's technique utilizing implicit integrators (Fatunla, 1988), and made crucial contributions to the development and application of block integrators (Fatunla, 1988). ...
In this paper, we developed a numerical scheme for modeled first order initial value problems in ordinary differential equations. We examined the formulation and application of a hybrid Block algorithm in this study. To develop a one-tenth-step continuous Block scheme, we utilized the collocation and interpolation of power series approximation strategy. Using the power series as the basis function and integrating it within one-tenth-step intervals we developed an alternative computational method that can be used to solve modeled first order initial value problems in ordinary differential equations. Additional investigation of the Block integrator's essential characteristics indicated that it is consistent, zero-stable, and convergent. The reliability and applicability of the technique are verified using numerical examples. The method is more accurate and algorithmically dependable than some approaches.
... Linear Multistep Methods (LMMs) for solving ODEs can be generated using methods such as Taylor's series, numerical integration and collocation methods, which are restricted by an assumed order of convergence [7]. Block integrators for solving ODEs have been developed by W. E. Milne [10] who employed them as starting values for predictor-corrector algorithms, [4] improved Milne's method in the form of implicit integrators and [9] also contributed greatly to the development and application of block integrators. Various authors [3], [1], [7], [8], and [11][12][13] proposed LMMs to generate numerical solution to (1). ...
... Definition 2 [9]: The block (8) is said to be zero stable, if the roots , = 1,2, … , of the characteristic polynomial ( ) defined by ( ) = det( − ) satisfies | | ≤ 1 and every root satisfying | | ≤ 1 have multiplicity not greater than the order of the differential equation. Furthermore, as ℎ → 0, ( ) = ( − 1) where μ is the order of the differential equation, d is the order of the matrix ( ) and E. For our method (13) This implies that our methods are zero-stable. ...
In this paper a numerical scheme has been developed for solving second order linear integro-differential equations subject to mixed conditions. We adopted the standard collocation points and the integro-differential equations is transformed into a system of linear equations. The linear system is then solved using MATLAB programming. The proposed scheme exhibits convergence, while the efficiency and applicability of the scheme has been demonstrated using two examples. The results were compared with an existing method that used Laguerre polynomials. The proposed method is computationally reliable.
... Linear Multistep Methods (LMMs) for solving ODEs can be generated using methods such as Taylor's series, numerical integration and collocation methods, which are restricted by an assumed order of convergence [7]. Block integrators for solving ODEs have been developed by W. E. Milne [10] who employed them as starting values for predictor-corrector algorithms, [4] improved Milne's method in the form of implicit integrators and [9] also contributed greatly to the development and application of block integrators. Various authors [3], [1], [7], [8], and [11][12][13] proposed LMMs to generate numerical solution to (1). ...
... Definition 2 [9]: The block (8) is said to be zero stable, if the roots , = 1,2, … , of the characteristic polynomial ( ) defined by ( ) = det( − ) satisfies | | ≤ 1 and every root satisfying | | ≤ 1 have multiplicity not greater than the order of the differential equation. Furthermore, as ℎ → 0, ( ) = ( − 1) where μ is the order of the differential equation, d is the order of the matrix ( ) and E. For our method (13) This implies that our methods are zero-stable. ...
In this paper, we considered the derivation and implementation of a hybrid Block method for the direct solution of first order
initial value problems in ordinary differential equations. We adopted the method of Collocation and Interpolation of power series
approximation to generate a one – twentieth step continuous Block scheme. We further investigate the key properties of the Block
integrator and we found it to be consistent, zero-stable and convergent. Numerical examples are presented to test the accuracy and
effectiveness of the method. The method is computationally reliable and more accurate than some existing methods.
... The system of IVPs (1) holds, [1], [2], [3]. ...
The quest for more efficient schemes that handle stiff initial value problems (IVPs) has led to the emergence of new families of methods. Amongst these methods is the Adaptive Backward Differentiation Formula (A-BDF), which is a variant of the Backward Differentiation Formula (BDF). In this study, a family of Block Adaptive Backward Differentiation Formulae is considered and it is developed through Taylor series expansion and the method of undetermined coefficients. The methods constructed herein are of order p = 2k. Numerical experiments on standard test problems show that the proposed methods produced more accurate solutions than existing methods in the literature.
... Following Fatunla [18], we associate a linear difference operator L[y(x); h] given by ...
In solving ordinary differential equations, tackling stiff problems necessitates the application of robust numerical methods endowed with A-stability properties. To circumvent the constraints posed by the Dahlquist barrier theorem and mitigate errors arising from step-by-step implementation of linear multistep methods, block hybrid schemes have been introduced. This study focuses on the development of novel block schemes designed for the direct approximation of solutions to stiff initial value problems. The methods proposed herein leverage both interpolation and collocation, enhancing their consistency, convergence, and accuracy in solving initial value problems. The efficacy of the devised methods is demonstrated through a comprehensive analysis of stability regions for each of the constructed block algorithms. Notably, these stability regions are proven to be unbounded for order . Comparative assessments reveal their competitiveness with existing methods. In fact, this research introduces innovative approaches to address the challenges posed by stiff initial value problems, offering enhanced stability and accuracy in comparison to established methods.
... Theorem 3.1 [5]: For a linear multistep method to be convergent, it is both necessary and sufficient for the method to be consistent and zero-stable. Therefore, the proposed block method is convergent, as it is both zero-stable and consistent. ...
This paper presents a novel fourth-order block method for the direct integration of second-order differential equations. The method is derived from a basis function that combines a third-order polynomial with the sum of sine and cosine functions. By leveraging this unique basis function, the proposed method maintains computational effi�ciency while achieving fourth-order accuracy. It outlines the method's derivation and analyzes its stability and accuracy properties. Numerical experiments demonstrate its e�ffectiveness and effi�ciency compared to existing techniques. The results indicate that the proposed fourth-order block method off�ers signi�cant advantages in accuracy and computational cost, making it promising for directly integrating
... Scholars have suggested various numerical methods to approximate initial value problems, spanning from discrete techniques (Lambert, 1973;Butcher, 2008;Fatunla, 1988), to methods employing prediction and correction (Kayode & Adeyeye, 2011;Adesanya et al., 2008;Awoyemi & Idowu, 2005) and subsequently, block techniques (Tumba et al., 2019;Sabo et al., 2019). Researchers have suggested various numerical techniques to solve equation (1.1), which can be categorized into single-step or multi-step methods. ...
This research delves into a comprehensive examination of the application and convergence analysis of a newly developed block method for simulating epidemic models. The focal point of this study revolves around the derivation and implementation of a novel scheme, crafted through the utilization of power series polynomials, ensuring the fulfilment of essential properties. The formulation of the new scheme was rooted in the power series polynomial, a mathematical construct known for its versatility and precision. The rigorous validation process confirmed that the derived scheme satisfied the requisite properties, thereby establishing its theoretical soundness. The crux of the investigation lies in the practical application of this innovative scheme to simulate an epidemic model. Through meticulous simulations, the results yielded compelling evidence of the new method's superiority over existing approaches considered in this research. The comparative analysis demonstrated a notable enhancement in both accuracy and convergence speed, highlighting the efficacy of the newly proposed scheme in capturing and predicting the dynamics of epidemics. The observed advantages of the new scheme are particularly noteworthy, showcasing its potential to revolutionize the field of epidemiological modelling. By outperforming established methods, the new approach not only contributes to the theoretical underpinnings of epidemic modelling but also holds significant promise for practical applications, such as forecasting disease spread and optimizing intervention strategies.
... In Eq. 10 and solving for R, the values of Rare obtained as 0 and 1. According to Simeon (1988Simeon ( , 1991, the block formulae represented by (Eq. 6) are zero-stable, since from Eq. 10, ρ (R ) = 0 satisfy |R|# 1, j = 1 and for those roots with |R j | = 1 the multiplicity does not exceed two. ...
A step hybrid block method for the solution of Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this study. The method is formulated from continuous schemes obtained via. collocation and interpolation techniques and applied in a block-by-block manner as numerical integrator for first, second and third order ODEs. The convergence properties of the method are discussed via. zero-stability and consistency. Numerical examples are included and comparisons are made with existing methods in the literature.
... Since the unit modulus is simple, then, BTFM is zero-stable. See ( [35] and [34] ). ...
A fifth-order block trigonometrically fitted method is considered for numerical solution of Volterra Integro-Differential Equations (VIDE). The proposed scheme is constructed using mul-tistep collocation technique. The fundamental properties such as zero-stability, and region of stability of the proposed scheme are established, while its performance is established through examples from recent scientific literature.
... First characteristic polynomial satisfies |λ| ≤ 1 and for the roots with |λ| = 1 , the multiplicity must not exceed the order of the differential equations according to Fatunla [29]. This kind of stability issue concerned with the behaviour of the different system when h→ 0. For h→ 0, the system of equations can be written as: ...
The application of block methods in the study of dynamical systems is an area that requires further research. Therefore, this work presents block methods and their application to solving some problems in solid and fluid mechanics. The method was developed directly using the approaches of collocation and interpolation, and utilizing power series polynomials as trial solutions. First, the system of linear equations is solved to obtain the unknown coefficients. These coefficients are then substituted into the approximate solution to obtain a continuous scheme. The continuous scheme and its first, second, and third derivatives are evaluated at all the grid points to generate the block methods. The derived methods were applied to solve fourth-order boundary value problems of ordinary differential equations arising from beams and chemical problems. The results demonstrate the reliability and efficiency of the proposed method.
... Our method is thus consistent since it is of uniform order 5. Note that consistency controls the magnitude of the local truncation error committed at each stage of the computation, Fatunla (1988). ...
In this research, we have proposed the numerical application of second derivative ordinary differential equations using power series for the direct solution of higher order initial value problems. The method was derived using power series, via interpolation and collocation procedure. The analysis of the method was studied, and it was found to be consistent, zero-stable and convergent. The derived method was able to solve highly stiff problems without converting to the equivalents system of first order ODEs. The generated results showed that the derived methods are notable better than those methods in literature. We further sketched the solution graph of our method and it is evident that the new method convergence toward the exact solution.
... The formulated model often yields an equation that contains the derivatives of an unknown function. Such an equation is referred to as Differential equation, (Fatunla, S. O.,1988). As opposed to one-step methods, which only utilize one previous value of the numerical solution to approximate the subsequent value, multistep methods approximate numerical values of the solution by referring to more than one previous value (Skwame, Y., Sabo, J. and Kyagya, T. Y., 2017). ...
This paper formulates an A-Stable uniform order six linear multi-step method for testing stiffly first order ordinary differential equations. Power series approximate solution was used as the basic function using interpolation and collocation techniques to derive the continuous schemes which was found to converge. The developed method is then applied to solve the system of first-order stiffly ordinary differential equations and the result is displace side by side in tables with the existing once, the method tend to give better approximation when implemented numerically than the existing method with which we compared our results.
... Kumleng, et al. [7] construct a family of continuous block A-stable third derivative for numerical integration of (1.1). Other authors who have done considerable work on the numerical solution of (1.1) case include [8,9,10], to predictor-corrector methods [11,12,13], and then block methods [14,15,16,17]. ...
The new accurate implicit quarter step first derivative blocks hybrid method for solving ordinary differential equations have been proposed in this paper via interpolation and collocation method for the solution of stiff ODEs. The analysis of the method was study and it was found to be consistent, convergent, zero-stability, We further compute the region of absolute stability region and it was found to be stable A . It is obvious that, the numerical experiments considered showed that the methods compete favorably with existing ones. Thus, the pair of numerical methods developed in this research is computationally reliable in solving first order initial value problems, as the results from numerical solutions of stiff ODEs shows that this method is superior and best to solve such problems as in tables and figures. Original Research Article Tumba et al.; ARJOM, 13(3): 1-13, 2019; Article no.ARJOM.47528 2
... The conventional approach for solving higher order initial value problems is by reducing them to their equivalent systems of first order ordinary differential equations and then solving by using the existing methods. This is widely discussed in Goult, et al. [8] , Lambert [9] , Lambert and Watson [10] , Fatunla [11] and Awoyemi [12] . However, this approach has some setbacks such as burden of computing and difficulties in designing computer programme which may have affect the accuracy of the method in terms of error. ...
The numerical application of third derivative on third order initial value problem of ordinary differential equations is consider in this paper. The method is derived by collocating and interpolating the approximate solution in power series, while Taylor series is used to generate the independent solution at selected grid and off grid points. The basic analysis of the method were established and it was found to be consistent, zero-stable and convergent. The developed method is then applied to solve some third order initial value problems of ODEs, and the result computed shows that the derived method is more accurate than some existing methods considered in this paper. We further plotted the solution graph of each problems and it is obvious that the numerical solution convergence toward the exact solution. Introduction Many problems in Physics, Chemistry, and Engineering science are demonstrated mathematically by third-order boundary value problems or initial value problems. These boundary value problems can be found in different areas of applied mathematics and physics as, in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves, or gravity driven flows. Third-order boundary value problems were discussed in many papers in recent years Ejaz and Mustafa [1] , Areo and Omojola [2] , Adeyeye and Omar [3] , Sunday [4] , Fsasis [5] , Omar and Adeyeye [6] , Awoyemi, Kayode and Adeghe [7] , etc. Ordinary differential equations are widely applied to model real life situations particularly involving engineering problems. The third order initial value problems of ordinary differential equations of the form
... Adopting the approach of Fatunla (1989), let the local truncation error (LTE) corresponding with (2) to be the linear difference operator such that Assuming that ( ) is sufficiently differentiable function, the terms in (12) is expanded through Taylor series about the point to have ℒ[ ( ); ℎ] = 0 ( ) + 1 ℎ ′ ( ) + 2 ℎ 2 ′′ ( ) + ⋯ + ℎ ( ) + ⋯ where, ...
... Following the approach of Fatunla [45] and Lambert [46], the continuous scheme (4) can be expressed in form of a linear difference operator ...
A class of high-order continuous extended linear multistep methods (HOCELMs) is proposed for solving systems of ordinary differential equations (ODEs). These continuous schemes are obtained through multistep collocation at various points to create a single block method with higher dimensions. This class of schemes consists of A-stable methods with a maximum order of , capable of yielding moderately accurate results for equations with several eigenvalues of the Jacobians located close to the imaginary axis. The results obtained from numerical experiments indicate that these schemes show great promise and competitiveness when compared to existing methods in the literature.
... Definition 3.2 (Fatunla, 1988) An algorithm is said to be zero-stable, if the roots ...
Third-order problems have been found to model real-life phenomena such as thick film fluid flow, boundary layer problems, and nonlinear Genesio problems, to name a few. This research focuses on the development of an algorithm using a basis function that combines an exponential function with a power series. This algorithm, called the Exponential Function Induced Algorithm (EFIA), was formulated using the collocation and interpolation techniques. The theoretical analysis of the algorithm was also carried out in this research. The outcome of the analysis showed that the algorithm is consistent, convergent, and zero-stable. The EFIA was applied to solving some real-life third-order problems like thin film flow and nonlinear Genesio problems. The numerical and graphical results obtained showed that the EFIA is accurate, has high precision, and is easy to implement. Algorithm, Exponentially-fitted function, Hybrid block method, Nonlinear Genesio equations, Thin film flow problem, Third-order
... However, to some extent, regression is still possible using SNNs as presented in [10]. Following regression, we have shown that SNNs can be used for operator regression, numerical schemes for dynamical systems [11], physics-informed machine learning [12], and more [13]- [15]. There is still a lot to study, revolving around neuromorphic algorithms for SciML, and this work presents a fundamental step. ...
This paper explores the impact of biologically plausible neuron models on the performance of Spiking Neural Networks (SNNs) for regression tasks. While SNNs are widely recognized for classification tasks, their application to Scientific Machine Learning and regression remains underexplored. We focus on the membrane component of SNNs, comparing four neuron models: Leaky Integrate-and-Fire, FitzHugh-Nagumo, Izhikevich, and Hodgkin-Huxley. We investigate their effect on SNN accuracy and efficiency for function regression tasks, by using Euler and Runge-Kutta 4th-order approximation schemes. We show how more biologically plausible neuron models improve the accuracy of SNNs while reducing the number of spikes in the system. The latter represents an energetic gain on actual neuromorphic chips since it directly reflects the amount of energy required for the computations.
... According to Lambert (1973) and Fatunla (1988) and , the ( New 4pb ) method is convergence since it is zero stable and consistent. ...
In this study, we devised 4-point implicit block (New 4pb) approaches that take advantage of second derivatives. The aim provides is more accurate and quicker numerical solutions to first - order ordinary differential equations (ODEs). Whereas the properties of the (New 4pb) method such as the order and zero-stability have been discussed. After that, the approaches are applied to a collection of first-order (ODEs). In addressing the set of test problems, numerical results clearly indicate that the newly proposed systems outperformed previous, well-known existing methods.
... Applications in some areas such as chemical reactions, control systems, and electronic networks lead to systems of stiff differential equations, they are stiff problems because, the solution being sought is varying slowly, but there are nearby solutions that vary rapidly, for such reason, the numerical method must take small steps in order to obtain satisfactory results. x [3]. Euler and RungeKutta method are some of the examples of one step method. ...
One-step second derivative block method with three off-step points is developed using Chebyshev collocation points for the solution of stiff IVPs of ODEs. The stability properties of the new method are tested and found to be of order five and convergent, and test the accuracy of the method on some numerical examples, results obtained show that this method is better in terms of accuracy when compared with the methods of the same order and higher order methods.
This study presents the construction of the four-point semi-implicit block methods with extra derivatives for
solving y′ = f(t, y). the proposed block methods are formulated using Hermite interpolating polynomials, which
approximate the solution of the problem at four points simultaneously. t e proposed block methods obtain the numerical
solutions directly without reducing the equation into the first-order system of ordinary differential equations. t e zerostability and convergent properties of the proposed methods are also investigated. n numerical results are presented, and
other block methods are compared. t e outcomes are very encouraging for the suggested approach. t e proposed method
produces more accurate numerical solutions to the test problems than the existing method, which are in good agreement
with analytical and methodological solutions. t e performance shows that the proposed methods are very efficient in
solving first-order ordinary differential equations.
Attracted by the important role of ordinary differential equations in many physical situations like engineering, control theory, biological, and economics, a three quarter - step block hybrid algorithm is constructed for the purpose of solving first-order initial value problems of ordinary differential equations (FIVPODEs). Our approach leverages on off grid points other than the usual whole step point methods constructed by many researchers; this is viewed as an important shift from the usual norm. Some of the advantages of hybrid methods is that they possess remarkably small error constants as observed in Table I among other advantages. Some problems solved by some existing methods are also solved using the constructed algorithm to demonstrate the simplicity, validity and applicability of the method. The results obtained revealed that the method is suitable for all forms of first-order initial value problems be it linear or non-linear ordinary differential equations.
This research paper succinctly explains the derivation of a numerical scheme from the Newton-Raphson’s iterative method. A juxtapose of this newly developed scheme with Newton-Raphson’s shows an improvement of finding better approximations to any root of a real number.
Attracted by the importance of ordinary differential equations in many physical situations like, engineering, business and health care in particular, an effective and successful numerical algorithm is needed in order to explain many of the ambiguities about the phenomena in many fields of human endeavor. In this study, an interpolation and collocation technique are adopted in deriving a Block Hybrid Algorithm (BHA) for the numerical solution of systems of first-order Initial Value Problems (IVPs). To derive the BHA, the shifted Legendre polynomials was interpolated at two selected points and its derivative was collocated at seven selected points. This led to a continuous scheme which was eventually evaluated at some points to obtain the discrete schemes used in the numerical computation. Furthermore, some illustrative examples are introduced to show the applicability and validity of the proposed algorithm. It was observed that the proposed algorithm has the desired rate of convergence to the exact solution. The suggested method utilizes data at points other than the step numbers which is viewed as an important landmark; another major advantage of this algorithm is that it possesses remarkably small error constants (Table 2). Some graphical representations of the exact and numerical results are presented to show how accurate the numerical results agree with the exact solutions.
In the era of computational advancements, harnessing computer algorithms for approximating solutions to differential equations has become indispensable for its unparalleled productivity. The numerical approximation of partial differential equation (PDE) models holds crucial significance in modelling physical systems, driving the necessity for robust methodol-ogies. In this article, we introduce the Implicit Six-Point Block Scheme (ISBS), employing a collocation approach for second-order numerical approximations of ordinary differential equations (ODEs) derived from one or two-dimensional physical systems. The methodology involves transforming the governing PDEs into a fully-fledged system of algebraic ordinary differential equations by employing ISBS to replace spatial derivatives while utilizing a central difference scheme for temporal or y-derivatives. In this report, the convergence properties of ISBS, aligning with the principles of multi-step methods, are rigorously analyzed. The numerical results obtained through ISBS demonstrate excellent agreement with theoretical solutions. Additionally, we compute absolute errors across various problem instances, showcasing the robustness and efficacy of ISBS in practical applications. Furthermore, we present a comprehensive comparative analysis with existing methodologies from recent literature , highlighting the superior performance of ISBS. Our findings are substantiated through illustrative tables and figures, underscoring the transformative potential of ISBS in advancing the numerical approximation of two-dimensional PDEs in physical systems.
The collocation and interpolation technique on the power series approximate solution was used to develop a new continuous implicit hybrid one-third step method capable of solving first-order stiff initial value problems. This method has the advantage of allowing for easy step length changes and function evaluation at off-step points. The block method used to implement the main method ensures that each discrete method resulting from the block's simultaneous solution has the same order of accuracy as the main method. The method provides large intervals of absolute stability; it is consistent, zero stable, and thus converges. Sample examples of linear and nonlinear stiff problems were used to evaluate the method's performance. When compared to other existing methods, our developed method has been shown to be more accurate.
In this paper, the development and analysis of a proposed method for solving physical models arising from real-life scenarios are presented. The proposed method is derived via the transcendental function of exponential type, examined and studied for its properties. In addition, the effectiveness of the method is evaluated by applying it to three numerical examples that originated from real-world scenarios. Moreover, this study presents a comparison of the outcomes generated by the proposed method and the existing method, in the context of the exact solution. The study concludes that the proposed method solves real life problems with the expected level of accuracy and, therefore, can be considered among the numerous methods that are appropriate and suitable for solving first-order initial value problems (IVPs).
The main aim of the article is to construct a new numerical scheme for the solution of domestic violence model. A comprehensive examination of the properties for the scheme shall be worked on. The proposed scheme was tested on domestic violence model that comprises of six state variables. The results generated by the scheme are analyzed and plotted.
We have proposed the direct solution of high order initial value problems of ordinary differential equations on one-step third derivative block method using power series method by interpolation and collocation in this research. Numerical analysis shows that the developed method is consistent, convergence and zero stable after testing the basic property. The new method was examined on three highly stiff numerical experiments and the results perform favorably than the existing method.
The development of numerical methods for the solution of initial value problems in ordinary differential equation have turned out to be a very rapid research area in recent decades due to the difficulties encountered in finding solutions to some mathematical models composed into differential equations from real life situations. Researchers have in recent times, used higher derivatives in the derivation of numerical methods to produce totally new ways of solving these equations. In this article, a new Runge-Kutta type methods with reduced number of function evaluations in the increment function is constructed, analyzed and implemented. This proposed method border on the use of higher derivatives up to the second derivative in the ki terms of Runge-Kutta method in order to achieve a higher order of accuracy. The qualitative features: local truncation error, consistency, convergence and stability of the new method were investigated and established. Numerical examples were also performed on some initial value problems to confirm the accuracy of the new method and compared with some existing methods of which the numerical results show that the new method competes favorably.
A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.
Fuzzy differential equation models are suitable where uncertainty exists for real-world phenomena. Numerical techniques are used to provide an approximate solution to these models in the absence of an exact solution. However, existing studies that have developed numerical techniques for solving FIVPs possess an absolute error accuracy that could be improved. This is as a result of the low order and non-self-starting properties of the developed numerical techniques by previous studies. For this reason, this study, develops an Obrechkoff-type two-step implicit block method with the presence of second and third derivative for the numerical solution of first-order nonlinear fuzzy initial value problems. The convergence properties for the proposed block method are described in detail. Then the proposed method is adopted to solve first-order nonlinear fuzzy initial value problems with triangular and trapezoidal fuzzy numbers. The obtained results indicates that the proposed method effectively solves first-order nonlinear fuzzy initial value problems with better accuracy.
In this study, we consider the system of second order nonlinear boundary value problems (BVPs). We focus on the numerical solutions of different types of nonlinear BVPs by Galerkin finite element method (GFEM). First of all, we originate the generalized formulation of GFEM for those type of problems. Then we determine the approximate solutions of a couple of second order nonlinear BVPs by GFEM. The approximate results are unfolded in tabuler form and portrayed graphically along with the exact solutions. Those results demonstrate the applicability, compatibility and accuracy of this scheme.
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