Numerical solutions of second-orderdifferential equationsby Adam Bashforth method

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1 American Journal of Engineering Research (AJER) 4 American Journal of Engineering Research (AJER) eissn : 3847 pissn : 3936 Volume3, Issue6, pp383 wwwajerorg Research Paper Open Access Numerical solutions of secondorderdifferential equationsby Adam Bashforth method Shaban Gholamtabar, Nouredin Parandin, Department of Mathematics, Islamic Azad University of Ayatollah Amoli Branch, Amol,Iran Department of Mathematics, Islamic Azad University of Kermanshah Branch, Kermanshah,Iran Abstract: So far, many methods have been presented to solve the firstorder differential equations Not many studies have been conducted for numerical solution of highorder differential equations In this research, we have applied Adam Bashforthmultistep methods to approximate higherorder differential equations To solve it, first we convert the equation to the firstorder differential equation by order reduction method Then we use a singlestep method such as Euler, Taylor or RungeKutta for approximation of initially orders which are required to start Adam Bashforth method Now we can use the proposed method to approximate rest of the points Finally, we examine the accuracy of method by presenting examples Keywords: highorder differential equations, Adams Bashforth multistep methods, order reduction I INTRODUCTION Differentialequations arevery useful indifferent sciencessuch asphysics, chemistry, biology and economy To learn more aboutthe use of theseequationsin mentioned sciences, you can see theapplication of theseequationsinphysics[,8,6,7,], chemistry in [6,,5], biology in [5,4] and economy in [] Considering that most of the time analyticsolution of such equations and finding an exact solution has either high complexity or cannot be solved, we applied numerical methods for the solution Due to our subject which is solving the secondorder differential equations, we will refer to some solution methods which have been proposed in recent years by other researchers to solve the equations In 993, Zhang presented a solution method for secondorder boundary value problems [] In, Yang introduced quasiapproximate periodic solutions for secondorder neutral delay differential equations[7] In 3, Yang presented a method for solving secondorder differential equations with almost periodic coefficients [8] In 5, Liu et alobtained periodic solutions for highorder delayedequations [9] In 5, Nieto and Lopez used Green s function to solve secondorder differential equations which boundary value is periodic [] In 6, Yang et al also applied Green s function to solve secondorder differential equations [9] In 8, Pan obtained periodic solutions for highorder differential equations with deviated argument [3] In, Lopez used nonlocal boundary value problems for solving secondorder functional differential equations [4] It should be noted that most of these equations have piecewisearguments Other parts of this paper are organized as follows In the second part, we willreview therequired definitionsandbasic concepts In the third section, we will propose the basic idea for solve insecond orderdifferential equations In thefourth section, we will provideexamplesfor further explanation ofthe method and in fifth section; it will end with results of discussion II REQUIRED DEFINITIONS AND BASIC CONCEPTS Definition [3]the general form of a secondorder differential equation is as follows: d y p(t) q(t)y R(t) dt dt Or it can be more simply stated as below: y p( t) y q( t) y R( t) 3 w w w a j e r o r g Page 38

2 American Journal of Engineering Research (AJER) 4 In which, p (t), q (t) and R (t) are functions of t and without any reduction in totality, the coefficient of y is equal to because also in another way, this coefficient will alrea convert to one for y with dividing by coefficient of y Note:Ifine theequation (3) the value ofr (t) is, then the equationiscalledahomogeneousequation Definition [3]we say that the function of with variable of y on series of is true in Lipshitz condition If a fix such exists with this property that when (3), We consider the fix L as a Lipschitz fix for Definition 3 [3]we say that the set of is convex If, whenever belongs to, the point of which belongs to per eachλ Theorem[3] Suppose that is described on a convex set of If a fix such exists that per each, Then according to the variable of y on in lipschitz condition is true with L fix lipschitz Theorem3 [3]suppose that and are contiguous on D when is true in lipschitz condition according to the variable of y on, then problem of initial value of, and have unique solution of per Definition 34 [3]a multisteps technique to solve the problem of initial value (33) It is a technique that its differential equation is to find the approximation of which can be shown by below equation in which is an integer greater in the network point of (34) Per each in which the initial values of (35) Are determined and as typical When, we call it explicit or open method and in the equation (3) it gives the value of explicitly based on predetermined values When, we call it implicit or close method, because appears in both sides of (34) and it can be determined only with an implicit method III THE MAIN IDEAFOR SOLVINGSECONDORDERDIFFERENTIAL EQUATIONS We consider the differential equations as following in which t is an independent variable and y is a dependant variable d (4) y = f(t, y, ) dt dt Now to solve such equations, we act as follows: (4) = p dt According to the equation (3), we convert the relation (3) to two firstorder differential equation as follows: (43) = p = f dt t, y, p dp (44) = p = f dt t, y, p Then we use Adam Bashforth s twosteps, threesteps and method to solve the equations (43) and (44) Suppose that the initial condition for (43) and (44) are given as below: (45) y t = y, y t = p t = p w w w a j e r o r g Page 39

3 American Journal of Engineering Research (AJER) 4 Now, if we want to use nstep Adam Bashforth method for n=, 3, 4 in this case we should use n initial step with a singlestep method such as Euler, Taylor or RungeKutta In this research, for example we describe the RungeKutta singlestep methodto approximate the n initial step as following: (46) y t m = y m, y t m = p t m = p m < m < n Then we describe k = hf t m, w m, v m (47) k = hf t m + h, w m + k, v m + l k = hf t m, w m, v m (48) k = hf t m + h, w m + k, v m + l Now suppose that we want to obtain the value ofv m +, w m + by v m, w m, we have: w m + = w m + k + k (49) v m+ = v m + l + l Now suppose that we are in n step that after this step we want to use Adam Bashforth multistep method; general form of these methods is as below: (4) w i+ = a n w i + a n w i + + a w i n + h b n f (t i, w i, v i + + b f t i n, w i n, v i n w = α, w = α,, w n = α n That the relation (3) is determined peri = n, n +,, N And also v i+ = c n v i + c n v i + + c v i n + h b n f (t i, w i v i (4) + + b f t i n, w i n, v i n v = β, v = β,, v n = β n Now in continue we will discuss about the process of obtaining multistep methods mentioned above briefly 5 Numerical examplesand tables In this section, the approximate solutions obtained from Adam Bashforth multistep methods are compared with exact solution by using numerical example and we obtain their proximity rate to the exact solution Example: find the solution of the following secondorder linear equation y 6y + 9y = y =, y =, t, The exact solution isy = te 3t Note :we have shown mstep Adam Bashforth methods and their derivatives with AB m و AB m respectively in below tables Note :in below tables, which we have used mstep Adam Bashforth methods for approximation, we have obtained previous m step with a singlestep method; here Euler and RungeKutta methods are applied to obtain previous m step t Table : in this table, the approximate solutions obtained from AB, are compared with each other to approximate the true answer y AB y AB y w w w a j e r o r g Page 3

4 American Journal of Engineering Research (AJER) 4 Table :in this table, the approximate solutions obtained from و AB methods are compared witheach other to approximate true answer derivation which meansy t Y AB y AB y Again we solve the previous example with twostep and threestep methods except that in previous examples we selected Euler driver singlestep method and tables 3 and 4 are RungeKuttadriver singlestep method Table 3: in this table, the approximate solutions obtained from و AB methods are compared with eachother to approximate the true solution (RungeKutta secondorder single step method is used) T Y AB y AB y Table 4: in this table, the approximate solutions obtained from و AB are compared with each other for derived approximation of true solution (RungeKutta secondorder single step method is used) T y AB y AB y IV CONCLUSION In this research, we used Adams Bashforth multistep methods to solve highorder differential equations In the example solved, tables and respectively are showing the comparison of approximate solutions obtained from multistep methods and their derivatives with true answer and derivatives of true answer Tables 3 and 4 are as well as tables and except that in tables and, Euler driver singlestep method is used and in tables 3 and 4, RungeKutta deriver singlestep method is used It should be noted that the accuracy of RungeKutta driver method is more than the Euler driver method However, in Adam Bashforth method, as the number of steps increase, the greater accuracy will be obtained ACKNOWLEDGEMENTS This article has resulted from the research project supported by Islamic Azad University of Ayatollah Amoli Branch in Iran w w w a j e r o r g Page 3

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