Abstract
In this paper, a two-dimensional Haar wavelet collocation method is applied to obtain the numerical solution of delay and neutral delay partial differential equations. Both linear and nonlinear problems can be solved using this method. Some benchmark test problems are given to verify the efficiency and accuracy of the aforesaid method. The results are compared with the exact solution and performance of the two-dimensional Haar collocation technique is measured by calculating the maximum absolute and root mean square errors for different numbers of grid points. The results are also compared with finite difference technique and one-dimensional Haar wavelet technique. The numerical results show that the two-dimensional Haar method is simply applicable, accurate and efficient.
1. Introduction
Delay differential equation (DDE)is a differential equation (DE) in which the time derivatives at the present time are dependent on the solution and possibly its derivatives at prior times. Time delay systems or prehistoric functions are words used to describe DDEs. DDEs are frequently also called time delay systems or prehistory functions. These equations have a wide range of applications in chemical processes, economics [1], ecology [2], engineering, medicine and other several fields [3]. The role of DDEs is prominent in modeling of various physical systems in natural and social sciences. From the last twenty years, researchers have taken interest in the solution of DDEs [4] and fractional DDEs. A Partial Differential Equation (PDE) containing one or more delay terms (means information about the prior time function to the present time of observation) is known as Delay Partial Differential Equation (DPDE). Time DPDE appears in neural network models, population dynamics, electrical systems and transport phenomena comprising line losing and conservative frameworks where the actions and impact are isolated by temporal intervals. DPDEs have numerous applications such as mechanics, fluid dynamics, diffusion process, economics, control theory and many other fields of science [5]. Due to its vast range of utilization in every discipline of science, it attracts the intentions of researchers to find the solution of DPDEs.
In literature some techniques are developed by researchers for the approximate solution of DPDEs and neutral DPDEs. The researchers concentrate on the uniqueness, existence, and stability of the analytical solution, which refers [6]. The numerical solution of a time-varying delay system by Chebyshev wavelets is presented by Ghasemi and Tavassoli in [7]. The author describes the technique depending on the expanding of various functions in the system as a truncated Chebyshev wavelet. The operational matrices of integration and delays are reduced to a system of algebraic equations. For the solution of the class of neutral delay parabolic DEs, Qifeng and Chengjian [8] used a compact difference scheme along with extrapolation techniques. Shakeri and Dehghan [9] develop a solution of DDEs through a homotopy perturbation technique.
Jackiewicz and Zubik [10] establish spectral collocation and waveform relaxation methods for nonlinear DPDE. In this method, waveform relaxation technique allows to replace the nonlinear DEs as a result of spectral collocation technique by a sequence of linear problems. Baleanu et al. [11] presents an approximation to the DDEs of noninteger order by the optimal method. Cimen and Yilmaz [12] study the numerical method for a neutral DDE. The authors present exponential basis functions and interpolation quadrature rules to generate a fitted difference technique on a uniform mesh, which is succeeded by the technique of integral identities. Barton et al. [13] develops a collocation scheme for periodic solution of neutral DDEs. This method depends on direct discretisation of the underlying neutral DDE and one based on a discretisation of a related differential difference equation. Raza and Khan [14] presents Haar wavelet series solutions for solving neutral DDEs. The author used the Haar wavelet to obtain the solution of neutral DDEs with respect to the mesh points. Vanani and Aminataei [15] presents the numerical solution of neutral DDEs utilizing the multiquadratic approximation scheme. Liu et al. [16] presents a differential transform method for some DDEs. Jin et al. [17] presents the neutral DPDEs with an implicit difference scheme and obtains accuracy of order two.
Singh et al. found the solution of nonlinear variable order fuzzy fractional PDEs [18], fractional model of telegraph equation [19], and fractional multidimensional diffusion equations [20]. Veeresha [21] demonstrates the chaotic behaviour of the coupled fractional-order system, Hammouch et al. [22] found solutions and synchronization of a variable-order fractional chaotic system. Yokus and Yavuz [23] gives comparison of numerical and analytical methods for fractional Burger–Fisher equation. Zada et al. [24] found approximate-analytical solutions to PDEs via auxiliary function method while Yavuz and Sene [25] found numerical solutions of the model describing fluid flow. Yavuz and Abdeljawad [26] found solutions of nonlinear fractional regularized long wave models.
Haar wavelet (HW) refers to the family of square integrable functions which are mathematically the foremost easy when compared with different wavelets. Small number of grid points gives high accuracy. HW technique is computer oriented technique, and it is easily implemented as compared to other wavelets. Because of these properties, HW technique is a better numerical technique for calculations and it is faster in data processing. HW has several applications in science and engineering and it is no surprise that the uses of HW for approximation purposes in the literature. The references [27–33] shows the work by using the HW method in the literature. The recent work using Haar technique for numerical solution of different problems in literature can be seen in [34–39]. Haar wavelets are piecewise constant functions with a jump discontinuity, and their derivatives vanish in all orders. As a result, these wavelets cannot be used to solve DPDEs and neutral DPDEs directly.
In this paper, 2D HW method is established for the numerical solution of nonlinear and linear DPDEs and neutral DPDEs. Assume DPDEs with the following delay in time: with delay and initial condition where , and boundary conditions , , where and are functions of time, , and are functions of .
Also, we will develop 2D HW technique for neutral DPDEs of the form [8]: with boundary conditions: and delay condition
where and are constant numbers.
The main aim of this article is to present a 2-dimensional HW scheme for numerical solution of DPDE and neutral DPDE of the types defined above. The highest space and time derivative showing in the Eq. (1) and Eq. (2) are approximated using 2-dimensional HW and the lower order space and time partial derivative and unknown function is acquired by the operation of integration. We acquire a system of algebraic equations by putting the collocation points by using the two-dimensional HW approach on Eqs. (1) and (2). For nonlinear systems Broyden’s method is used while Gauss elimination technique is utilized for the linear system to find the Haar coefficients. Finally, the numerical solution at the collocation points is acquired by utilizing these Haar coefficients .
The arrangement of the paper as: definition of HW is specified in Section 2. The numerical technique for the solution of DPDEs and neutral DPDE depends on two-dimensional HW is established in Section 3. Test problems are given in Section 4 and conclusion is given in Section 5.
2. Haar Wavelet
The scaling function on interval is prescribed as [33].
The mother wavelet on is
The family of the HW is defined as, where
where , where and the range is . The number can be related by the relation . , and in [0,1] are:
Any function in can be written as:
This series is abbreviated at finite terms as
We utilize the following symbol
and the integral involved in Eq.(12) is calculated by using definition of as
Generally
3. Numerical Method
In this section, numerical technique will be establish utilizing the 2-dimensional HW technique for solution of Eq. (1) and Eq. (2). The solution of both linear and nonlinear DPDEs will be studied in this section. Introduce some notation .
Consider a 2D Haar wavelet expansion to approximate the second order space derivative as follows: integrating and using the given conditions, we obtain
Again integrating from to , we get
Thus Eq. (15) becomes
Again integrating, we obtain the required solution
Time derivative approximation.
The time derivative is approximated as:
Integrating and using the given condition, we obtain the required solution
The numerical scheme for DPDEs and neutral DPDEs are discussed separately.
3.1. Scheme for DPDEs
In this subsection, we will find the solution of DPDE defined in Eq. (1). By using the boundary conditions and above approximation, the Eq. (1) becomes
After simplification, we obtain
The collocation points and , for , are
Discretizing at collocation points and , we have
Comparing Eq.(21) and Eq.(19), we obtain
Utilizing boundary condition and after simplification, we have
Discretizing at nodal points (24), we get
Combining Eq. (25) and Eq. (28) we acquire a system of linear equations of size . Gauss elimination method is utilized for numerical solution of the above system, to find the values of unknown Haar coefficients. The required solution is easily obtained by putting these unknowns and in Eq. (19) or (21).
3.2. Scheme for Neutral DPDEs
In this subsection, neutral DPDE given in Eq. (2) is considered. By using the above approximations and putting the values in Eq. (2), we obtain
Comparing Eqs. (19) and (21), we have
Combining Eq. (29) and Eq. (30), we obtain a simultaneous system of linear equations of order . Gauss elimination technique is utilized for the solution of the above system, to find the values of unknowns and . The required solution easily can be obtained by putting these unknowns and in Eq. (19) or (21).
4. Test Problems
Some numerical examples are considered in this section to exhibit the accuracy of the developed method. The maximum absolute errors at mesh points is denoted by . If represents the approximate and the exact solution at collocation points, then Root mean square (RMS) error at collocation points is defined as
The results are also compared with 1D HW technique and finite difference technique.
Example 1. First order PDEs with delay in time and deceleration of a state variable are called advection equations. Assume the advection equation with time delay [40]: where initial and delay conditions: The exact solution is The comparison of the errors for 2D HW technique with finite difference scheme [5] and one dimensional HW technique [33] are given in Table 1. A good performance of 2D HW technique is seen from this table. The errors at time using 2D HW technique for distinct numbers of grid points are given in Table 1. From Table 1, we examine that the errors are decreased by rising the number of grid points. This represents that the 2D HW technique has coherent behavior for this problem. The errors by Hybrid technique (HW and finite difference) are decreased to while by 2D HW technique is decreased up to . This shows that the 2D HW technique is more accurate than the finite difference scheme [40] and 1D HW technique [33] for this problem. We have shown the approximate solution at the collocation points for in Figure 1.
Example 2. Consider nonlinear DPDE [40]: with and delay condition: The problem have analytical solution
This is a nonlinear DPDE whose nonlinearity can be found in the delay term. So this can be solved by utilizing the same method of linear problem. The results for the distinct number of mesh points are given in Table 2. The numerical results show that the 2D HW technique has a good performance for this test problem. We can see that a good accuracy can be achieved by raising the mesh points. The results show that an accuracy can be obtained, which represents a better performance of the newly developed 2D HW technique as shown in Figure 2.
Example 3. Consider neutral DPDEs [8]. with delay condition: The exact solution is The RMS and errors for different numbers of nodal points are given in Table 3. The capability of the develop technique for this kind of problem is uniformly better which can comfortably be examined from Figure 3, also we see that if we extend the number of nodal points then and RMS errors are decreased.
Example 4. Consider the following neutral DPDE [8]. with: delay condition: , and . The analytical solution is . The RMS and errors for distinct numbers of nodal points are given in Table 4. From this table we examine that the RMS error decreased up to and the error decreased up to order for numbers of nodal points. If we extend the number of mesh points then RMS and errors are decreased. The surface plot of the approximate solution at is given in Figure 4.
5. Conclusion
A new numerical technique is established utilizing 2-dimensional HW for the solution of DPDEs and neutral DPDEs. Both the space and time derivatives are approximated by Haar functions. Some numerical examples are acquired to verify the convergence of the 2-dimensional HW method. The numerical results represent that the method is accurate and efficient. The technique is appropriate to both linear and nonlinear problems of DPDEs and neutral DPDEs. The results are also compared with a finite difference method and 1-dimensional HW technique. The results show that 2-dimensional HW technique is better than finite difference technique and 1-dimensional HW technique. The figures show the comparison of exact and approximate solution. The Haar technique can be applied to the system of integro-partial differential equations, system of delay PDEs and system of neutral delay PDEs.
Data Availability
The data used in this research is included within the paper.
Conflicts of Interest
No conflict of interest exists.
Authors’ Contributions
An equal contribution has been made by all the authors.
Acknowledgments
This research was funded by National Science, Research and Innovation Fund (NSRF), and King Mongkut’s University of Technology North Bangkok with Contract no. KMUTNB-FF-65-49.