Novel Schemes for Cauchy-Riemann System of Equations with Cauchy Conditions

This communication deals with the analytical solutions of Cauchy problem for Cauchy-Riemann system of equations which is basically unstable according to Hadamard but its solution exists if its initial data is analytic. Here, we used the Vectorial Adomian Decomposition (VAD) method, Vectorial Variational Iteration (VVI) method, and Vectorial Modi ed Picard's Method (VMP) method to get the convergent series solution. These suggested schemes give analytical approximation in an in nite series form without using discretization. These methods are e ectual and reliable which is demonstrated through six model problems having variety of source terms and analytic initial data.


Introduction
Many physical systems are used to predict the behavior of dierent phenomenon which are modeled in the form of partial dierential equations. In order to understand the underlying physics of these phenomena, solutions of these expressions is a matter of great importance. Researchers have proposed several numerical and analytical methods to obtain the solutions of complex systems. For instance, Santra et al. [1] presented the oscillation theorem and also discussed the consistency analysis of second-order dierential equations in their exploration. They proved several theorems for the validity of the solutions of this class of dierential system. Also, they established the condition for the oscillation of solution. Ahmed et al. [2] studied the KdV equations via dierent well known approaches. They have analyzed and noticed the behavior of solution by dierent methods and presented the comparative analysis by computing the error tables. Also, they plotted the behavior of exact and approximate solutions. Moreover, they presented the absolute error analysis for the authenticity and applicability of utilized method. Turkyilmazoglu [3] presented the modication in variational iteration scheme by introducing the concept of accelerating convergence parameter. Oil diusion model was studied by Ahmad et al. [4]. They presented the analytical solutions. They computed the residual errors and noticed that error diminish with the higher order approximation. Turkyilmazoglu [5] introduced the modication in Adomian scheme by using the convergence parameter. From several modeled examples, it is tested that the proposed modication increases the convergence rate as compared with the traditional approach. Several other important contributions can be found in [6]- [12].
In this article, we deal with the rst-order system of linear partial dierential equations [13] ∂ ∂y with Cauchy condition where φ(x) and ψ(x) are analytic and taking y as a time variable for the desired solution vector for real-valued functions u(x, y) and v(x, y). Collectively (1) is elliptic while individually both the partial dierential equations [14] are hyperbolic. According to Hadamard [15], the Cauchy problem for the CRE is ill-posed. Tikhonov [16] and Bertero et al. [17] also discussed the ill-posed problems and their solution.
Walter [18] proved the Cauchy-Kowalewsky theorem which guarantees existence and uniqueness of the Cauchy problem which are Hadamard unstable. The main source of motivation to our present article is Joseph et al. [19], according to them, the problems which are Hadamard unstable cannot be solved unless the initial data are analytic. Available literature witnessed that no study has been conducted to investigate the IVP for CR-equation in three dimensions. The next project will covers this type of problem in open literature.

Adomian's decomposition method
George Adomian [20]- [22] introduced a powerful technique for solving functional equations of any kind and stochastic problems which is known as the Adomian's Decomposition Method (ADM). This method provides an eective procedure for analytical solution of real physical problems. Its convergence is discussed by Abbaoui et al. [23], Abdelrazec et al. [24], and Turkyilmazoglu [25]. The ADM is used to tackle the problem directly without using linearization, perturbation, or any other restrictive assumptions.

Analysis of the Adomian's Decomposition Method
Let us consider the equation where F is a dierential operator involving both linear and non-linear terms. We write the Eq. (3) in operator form as where L = Invertible Highest order linear derivative, R = Contain the rest of the linear operators, N = The non-linear terms if there are any, g = The source term. Solving Eq. (4) for Lu, we have The inverse of L is L −1 , which is considered as denite integral, i.e. if L = d dx , then If L is second derivative, then L −1 can be dened as a two-fold integral. Applying L −1 on both sides of the Eq. (5) we have where f consist of integration of the source term and Cauchy conditions. Finally, the solution u is represented as the innite series The terms u 0 , u 1 , u 2 , . . . .. are obtained recursively through Eq. (11) and non-linear term N u is disintegrated as follows where A n are Adomian polynomial in u i , i = 0, 1, 2, 3, . . . . and generated by the formula Then calculating the components u 0 , u 1 , u 2 , . . . .. from the following relation recursively and putting these terms in Eq. (8), we get the required Adomian's Decomposition solution. So once the term u 0 is dened then the remaining terms u k , k ≥ 1 are completely determined.

Analysis of Vectorial Adomian's Decomposition (VAD) Method for Cauchy-Riemann Equations
To demonstrate the VADM, we consider with the given Cauchy conditions where invertible operator of highest order L y is given by L y = d/dy, and L x = d/dx.
f (x, y) and g (x, y) are the forcing terms. The integral operator L y −1 is dened by Applying L y −1 on both sides of Eq. (12), and using the given conditions, we obtain The function f (x, y) and g(x, y) consist of integration of the source term and all prescribed Cauchy So the series solution is determined as

Implementation of VAD Method
The proposed method is applied on the Cauchy problems for the Cauchy-Riemann systems of equations.

Model Problem-A
Consider the Cauchy-Riemann system of equation with Cauchy data and the exact solution is Applying VAD procedure on Eqs. (17)-(18) which gives the recurrence relation and Eq. (20) yields the following vectors Hence the solution in closed form is which is the same as exact solution.
Comparison of exact and approximated solution via VADM is given by the following graph

Model Problem-B
Consider the Eq. (17) subject to the initial data with the exact solution The recurrence relation is given as Thus we obtained the following vectors So we have which is the same as exact solution.
Comparison of exact and approximated solution is given by the following graph with the Cauchy conditions With the exact solution The recurrence relation is Thus we obtained the following vectors with the Cauchy conditions whose exact solution is The recurrence relation is given by Thus we obtained the following vectors with the homogeneous Cauchy conditions having exact solution is The recurrence relation is given by Thus we obtained the following vectors Compare the result by making graph of exact and approximate solution by subject to the homogeneous Cauchy conditions the exact solution is The recurrence relation (12) for (41)-(42) is given by Thus we obtained the following vectors

Vectorial Veriational Iteration (VVI) Method
For the solution of system of dierential equations many analytical and numerical methods have been developed. We consider the Eq. (4) and construct the correction functional for VIM as u n+1 = u n + t 0 λ (Lu n + Ru n + Nũ n − g) ds, where the Lagrange multiplier λ can be identied via variational theory, u n is the approximate value, andũ n denotes the restricted variation, i.e. δũ n = 0.
where the Lagrange multiplier λ can be identied optimally via variational theory. The Lagrange multiplier in the case of Cauchy-Riemann equations is -1 because this is 1st order linear system of equations.
Consequently, the solution is given by

Implementation of VVI Method
Some modal problems are presented here for the suitability and compatibility of the Vectorial Veriational Iteration Method (VVIM).
In closed form, we have The graphical comparison of exact and 10 th iterates of approximate solution with y=1 is given as Figure 13: Graph of exact solution and 10th iteration of u by taking y=1.

Model Problem-B
Consider the Eqs. (17) and (23) whose correction functional is which yields the following vectors By using the (51) we have The graphical comparison of exact and 10 th iterates of approximate solution with y = 1 is given as Figure 15: Graph of exact solution and 10th iteration of u by taking y=1.
The correction functional Eq. (56) yield the following vectors

Model Problem-D
Consider the inhomogeneous CRE with source terms and the initial data as trigonometric functions. The correction functional is The correction functional Eq. (57) yields the following vectors The comparison of exact and 10 th iterates of approximate solution when a = 2, b = 5 and y = 1 is expressed as Figure 19: Graph of exact solution and 10th iteration of u by taking y=1, a=2, and b=5.
The correction functional Eq. (59) yields the following vectors The graphical comparison of exact and 10 th iterates of approximate solution with y = 1 is given as Figure 21: Graph of exact solution and 10th iteration of u by taking y=1. ds.
The graphical comparison of exact and 10 th iterates of approximate solution with y = 1 is given as Figure 23: Graph of exact solution and 10th iteration of u by taking y=1.

(64)
They also showed that the nth Picard iterates is the Maclaurin polynomial of degree n for y(t) if φ n (t) is truncated to degree n at each step. This form of Picard's method is called the Modied Picard Method (MPM).

Analysis of the Vectorial Modied Picard's Method (VMPM) for Cauchy-Riemann Equations
We consider the Eq. (4) and applying the VMPM where n is the order of the approximation. The solution is, therefore, given by

Implementation of the Method
To check that the proposed method is eective for the initial data we solve six model problems.
This is the same as exact solution.
The graphical comparison of exact and 10 th iterates of approximate solution with y = 1 is given as

Model Problem-B
Consider the problem given in Eqs. (17) and (23) and handling it with the iterative relation (65), we , . . .
This is the same as exact solution.
The graphical comparison of exact and 10th iterates of approximate