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

Abstract

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 Modified Picard’s Method (VMP) method to get the convergent series solution. These suggested schemes give analytical approximation in an infinite series form without using discretization. These methods are effectual and reliable which is demonstrated through six model problems having variety of source terms and analytic initial data.

Keywords

• Cauchy-Riemann system of equations;
• Cauchy problem;
• Vectorial Adomian Decomposition method
• Vectorial Variational Iteration method
• Vectorial Modified Picard’s Method.

Kaynakça

1. [1] S.S. Santra, O. Bazighifan, H. Ahmad, and Yu-Ming Chu, Second-order differential equation: oscillation theorems and applications, Mathematical Problems in Engineering 2020 (2020).
2. [2] H. Ahmad, T. A. Khan, and Shao-Wen Yao, An e?cient approach for the numerical solution of fifth-order KdV equations, Open Mathematics 18, no. 1 738-748 (2020).
3. [3] M. Turkyilmazoglu, An optimal variational iteration method. Applied Mathematics Letters 24, no. 5 762-765 (2011).
4. [4] H. Ahmad, T. A. Khan, H. Durur, G. M. Ismail, and A. Yokus, Analytic approximate solutions of diffusion equations arising in oil pollution, Journal of Ocean Engineering and Science (2020).
5. [5] M. Turkyilmazoglu, Parametrized adomian decomposition method with optimum convergence. ACM Transactions on Mod- eling and Computer Simulation (TOMACS) 27, no. 4 1-22 (2017).
6. [6] U. Ali, M. Sohail, and F. A. Abdullah, An efficient numerical scheme for variable-order fractional sub-di?usion equation, Symmetry 12, no. 9 1437 (2020).
7. [7] M. Turkyilmazoglu, A reliable convergent Adomian decomposition method for heat transfer through extended surfaces, International Journal of Numerical Methods for Heat & Fluid Flow (2018).
8. [8] U. Ali, M. Sohail, M. Usman, F. A. Abdullah, I. Khan, and K. S. Nisar, Fourth-Order Difference Approximation for Time-Fractional Modified Sub-Diffusion Equation, Symmetry 12, no. 5 691 (2020).

9. [9] H. Ahmad, R. A. Seadawy, and T. A. Khan, Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm, Mathematics and Computers in Simulation (2020).
10. [10] T. Naseem, and M. Tahir, A novel technique for solving Cauchy problem for the third-order linear dispersive partial differential equation, International Journal of Physical Sciences 8, no. 6 210-214 (2013).
11. [11] H. Ahmad, R. A. Seadawy, T. A. Khan, and Phatiphat Thounthong, Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations, Journal of Taibah University for Science 14, no. 1 346-358 (2020).
12. [12] M. Turkyilmazoglu, A simple algorithm for high order Newton iteration formulae and some new variants, Hacettepe Journal of Mathematics and Statistics 49, no. 1 425-438 (2020).
13. [13] T. Naseem, and M. Tahir, Vectorial reduced differential transform (VRDT) method for the solution of inhomogeneous Cauchy-Riemann system, International Journal of Physical Sciences 9, no. 2 20-25 (2014).
14. [14] W.L Wendland, Elliptic systems in the plane (Vol. 3), Pitman Publishing (1979).
15. [15] J.Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations Yale University press, New Haven, (1923).
16. [16] A. N. Tikhonov, On the solution of ill-posed problems and the method of regularization. In Doklady Akademii Nauk, vol. 151, no. 3, pp. 501-504. Russian Academy of Sciences,(1963).
17. [17] M. Bertero, T. A. Poggio, and V. Torre, Ill-posed problems in early vision, Proceedings of the IEEE 76, no. 8 869-889 (1988).
18. [18] W. Walter, An elementary proof of the Cauchy-Kowalevsky theorem, The American mathematical monthly 92, no. 2 115-126 (1985).
19. [19] Joseph, D. Daniel, and J. C. Saut, Short-wave instabilities and ill-posed initial-value problems, Theoretical and Computa- tional Fluid Dynamics 1, no. 4 191-227 (1990).
20. [20] G. Adomian, A review of the decomposition method in applied mathematics, Journal of mathematical analysis and appli- cations 135, no. 2 501-544 (1988).
21. [21] G. Adomian, Nonlinear stochastic systems theory and applications to physics (Vol. 46), Springer Science & Business Media (1988).
22. [22] G. Adomian, Solving frontier problems of physics: the decomposition method (Vol. 60), Springer Science & Business Media (2013).
23. [23] K. Abbaoui, and Y. Cherruault, New ideas for proving convergence of decomposition methods, Computers & Mathematics with Applications 29, no. 7 103-108 (1995).
24. [24] A. Abdelrazec, and D. Pelinovsky, Convergence of the Adomian decomposition method for initial-value problems, Numerical Methods for Partial Differential Equations 27, no. 4 749-766 (2011).
25. [25] M. Turkyilmazoglu, Accelerating the convergence of Adomian decomposition method (ADM), Journal of Computational Science 31 54-59 (2019).