Abstract
In this research, a new approach is presented for solving delay differential equations (DDEs) which is a blend of Sumudu transform and variational iteration method (VIM). A general Lagrange multiplier is used to construct a correction functional. This is done with an uncommon Sumudu transform alongside variational theory. A few numerical cases were solved to demonstrate methodology of this new approach. Objective of this research is to reduce the complexity of computational work compared to the conventional approaches. It can be concluded that the amount of evaluation is reduced but at the same time the results are comparable as in the previous works.
1. Introduction
In recent years, especially, for several decades, applications of powerful tools for solving numerical and analytical cases have attracted attentions of scientists from all over the world [1]. Delay differential equations (DDEs) are a kind of functional differential equation having a widespread range of uses in the arena of science, technology, and engineering which acquires numerical/analytical solutions. It had already been applied in control theory for many years and recently it is being used extensively in many biological models. DDEs are form of differential equation in which the derivative of the unknown function at a particular period is provided in terms of values of the function at earlier periods. Introducing delays in models enhances the vitality of these models and allows an accurate depiction of actual occurrences. DDEs arise commonly in various physical occurrences. To be specific, they are essential once ordinary differential equation (ODE) based models are unsuccessful. In disparate ODEs, where initial conditions are stated at preliminary point, DDEs need history of the system over the delayed interval which are then provided as initial conditions. In line with this, delay systems turn out to be intricate and multifaceted in nature which in turn complicates the process of analyzing DDEs analytically and therefore requires a numerical approach. The delay differential equation in its simplest form isin which and is a constant delay.
Application of a few numerical methods that were introduced turns out to be very useful to solve these types of equations as in numerical simulations such as the variational iteration method and the optimal perturbation method. DDE problems have constantly led to an infinite spectrum of frequencies. Hence, few approaches have been implemented in solving them such as approximation, asymptotic solutions, numerical method, and graphical approaches. In recent times, Anakira et al. employed the Optimal Homotopy Asymptotic Method (OHAM) in solving linear and nonlinear DDE [2] while Alomari et al. used the Homotopy Analysis Method (HAM) to find the solution for DDE [3]. Among all these methods applied to solve differential equations, one of them is the variational iteration method (VIM) which was firstly initiated by Ji-Huan He which could be seen throughout [4–8]. Subsequent works [9–17] reflect the flexibility, consistency, and effectiveness of the procedure in VIM. The application of VIM to differential equations generally involves discovering a correction functional, finding the Lagrange multiplier and deciding on a good initial approximation. Application of VIM has remained successful on initial as well as boundary value problems, Schrodinger equations, integrodifferential equation, fractional differential equation, chaotic Chen system, coupled sine-Gordon equation, general Riccati differential equations, fractional heat- and wave-like equations, and many more.
Various authors have made efforts in the developments of the VIM with the wide application of the method. Particularly, Wu [13, 16] used the Laplace transform to compute the Lagrange multipliers, which overcomes main shortcomings in implementation of the VIM to fractional equations. Sumudu transform is a simple variant of the Laplace transform and is essentially identical with the Laplace. Sumudu transform has many interesting properties that make it easy to visualize making it an ideal transform for control engineering and applied mathematicians. Recently, Sumudu transform is implemented in some well-known analytical approaches [19], where the coupling of homotopy perturbation method (HPM) and Sumudu transform is used to make the process of the solution simpler and improves the solution’s accurateness.
A new modified variational iteration method was found, inspired and driven by Wu’s thoughts and combining with the Sumudu transform [20]. The new approach is based on variational iteration theory and Sumudu transform. In this paper, the basic motivation is extending this new reliable approach for the solution of linear and nonlinear DDEs which are normally challenging to analyze due to their multifaceted nature and boundless dimensionality.
2. Sumudu Variational Iteration Method (SVIM)
First, let us take the general nonlinear differential equation [21] to illustrate main idea of VIM for DDEs,with the following initial conditions:in which , is a linear operator, is a nonlinear operator, is a knowncontinuous function, and is the term of the maximum order derivative.
Basic idea of the VIM is building a correction functional for (2) of formula The successive approximation , could be attained by finding , a general Lagrange multiplier that could be known optimally with variational theory. The function is considered as a restricted variation indicating . At first, integration by parts is done to determine the Lagrange multiplier which enables the consecutive approximations, of the exact solution to be attained by means of a good initial approximation Initial conditions in (3) typically give the initial approximation. Finally, as converges to the exact solution
2.1. Combination of VIM and Sumudu Transform (SVIM)
The entire procedure of Lagrange multipliers is expressed as a case of algebraic equation where its solution could be found by Optimality condition for the extreme advances toin which represents the traditional variational operator. Implementing the initial point provided, the approximate solution could be determined via next iterative scheme, with (5) and (6)The formula above (7) is the famous Newton-Raphson formula that possesses a quadratic convergence.
In this article, we outspread the idea in finding the unknown Lagrange multiplier. Key step is the application of Sumudu transform into (2) with its fundamental properties in [22]. Then, the linear equation will be converted into an algebraic equation below: Hence, the algorithm of the Sumudu Variational Iteration Method (SVIM) is given below:(1)Applying Sumudu transform to (2) gives the correction functional as with the notation used to indicate Sumudu transform.(2)Considering the terms as restricted variations, we let (9) be stationary with respect to From (9), we define Lagrange multiplier as (3)Succeeding approximations can then be attained with the application of inverse Sumudu transform into (8) which gives
with initial approximationEquation (12) shows that the first iteration in traditional VIM is made up by the Taylor series.
3. Numerical Applications
Over this segment, application of the above procedure is done to some nonlinear DDEs.
Example 1. Consider a first order nonlinear DDE The exact solution is known as
Taking the Sumudu transform, we obtain The iteration formula thus isand its Lagrange multiplierApplying inverse Sumudu transform givesTherefore,with initial approximation and applying the iteration formula (19) above, we attainFrom these approximations, it can be seen that the solution tends to form the Taylor series expansion of . In order to attest numerically whether or not the suggested approach maintains the accurateness, numerical solutions of the approximation up to were evaluated. The absolute values of LVIM and SVIM are compared in Table 1 while Figure 1 displays behavior of the error between these two methods in Example 1. The outcome is in good agreement with each other. Terms of sequences obtained from SVIM are computed using the Maple package.
Example 2. Let us look at second order linear DDE for a second example.The exact solution is given by
Taking the Sumudu transform, we obtain and its Lagrange multiplierApplying inverse Sumudu transform giveswith initial approximation Thus, from (24), we attainThe convergence of the approximations to the exact solution could be observed from the solution. The absolute values of LVIM and SVIM are compared in Table 2 while Figure 2 displays behavior of error between these two methods in Example 2. The values demonstrate that the outcomes are in excellent agreement with those of the other approaches.
Example 3. Lastly, a third order nonlinear DDE is consideredExact solution is known as
Succeeding the processes of earlier examples, we attain the following consecutive approximations:That converges to exact solution when . Approximate solutions obtained with SVIM could be seen to have a good agreement with the other method. The absolute values of LVIM and SVIM are compared in Table 3 while Figure 3 displays the behavior of the error among these two methods in Example 3.
4. Conclusion
The Sumudu transform is a simple variant of the Laplace transform and is essentially identical with the Laplace. It has many interesting properties that make it easy to visualize and hence making the process of the solution simpler. Sumudu-Lagrange multiplier is derived from Sumudu transform plus integrating with procedures of VIM to obtain the approximate solutions to DDEs. Lagrange multipliers, which are defined in (11), could be known optimally with this different approach of variational theory. A new modification of the VIM was attained. This recommended procedure was obtained by not having applied any linearization, discretization, or impractical rules. This new technique provides more convincing or accurate sequence of results that converges quickly in physical problems. In this article, the SVIM was successfully applied in solving linear and nonlinear DDEs. In order to attest numerically whether or not the suggested approach maintains the accurateness, numerical solutions of the approximation up to were evaluated. The absolute values of LVIM and SVIM are compared in the tables while Figures 1, 2, and 3 display behavior of the error between these two methods in Example 1 to Example 3. The outcomes are in good agreement with each other. It is worth stating that this method is efficacious in minimizing number of computations contrasting with traditional methods in spite of sustaining great accurateness of the numerical outcome.
Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.