 Research note
 Open Access
 Published:
On solving initial value problems for partial differential equations in maple
BMC Research Notes volume 14, Article number: 307 (2021)
Abstract
Objectives
In this paper, we present and employ symbolic Maple software algorithm for solving initial value problems (IVPs) of partial differential equations (PDEs). From the literature, the proposed algorithm exhibited a great significant in solving partial differential equation arises in applied sciences and engineering.
Results
The implementation include computing partial differential operator (PartialDiffOperator), Greens function (GreensFunction) and exact solution (ExactSolution) of the given IVP. We also present syntax, ApplyPartialDiffOp, to apply the partial differential operator to verify the solution of the given IVP obtained from ExactSolution. Sample computations are presented to illustrate the maple implementation.
Introduction
In the scientific area, symbolic computation is one of the significant subjects, and it is playing dominant role in solving many mathematical equations, particularly the applications related to differential equations. In the symbolic computation research, the biggest success is the developing several substantial software systems. Several symbolic algorithms/methods have been created by various scientists, researchers and engineers; see, for example, [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. Most of these algorithms have been implemented in various mathematical software. Hence, in this paper, we focus on the implementation of the efficient algorithm presented in [13]. For more understanding and application on Maple software commands on how to obtain numerical solutions and the plots see [27,28,29,30,31].
In this paper, we mainly focused on the Maple implementation of the initial value problems (IVPs) for solving partial differential equations (PDEs) with constant coefficients. The proposed algorithm/method was introduced by S. Thota and S. D. Kumar in 2020, see [13] for more details. In [13], they presented a new symbolic method/algorithm to find the Green’s function for a given IVP of linear second order PDEs with constant coefficients. In this method, they focused on computing the Green’s function using the integrodifferential algebras [1, 3, 9, 25]. They have discussed several numerical examples to illustrate the symbolic algorithm. They also briefly discussed about the implementation in Maple. The rest of paper is planned as follows. In Sect. 1.1, we briefly state the symbolic algorithm of IVP for PDEs, Sect. 2 presents the Maple implementation of the algorithm with pseudocode and Maple programming and Sect. 3 focused on sample computations to illustrate the implementation.
Symbolic method/algorithm
In this section, we recall the symbolic method/algorithm for IVPs for second order PDEs, see [13], for more details about the algorithm.
The general form of the IVP for a second order PDE with inhomogeneous initial conditions over \(\mathcal {F} = C^\infty (\mathbb {R}^2)\) is as follows.
where \(u(x,y) \in \mathcal {F}\) is unknown function which is going to be determined, called solution of (1), using forcing function \(f(x,y) \in \mathcal {F}\) and the initial data \(\alpha _1(y),\alpha _2(y) \in \mathcal {F}\). The proposed algorithm solves the IVP (1) not only for static values of f(x, y), \(\alpha _1(y),\alpha _2(y)\) but give a standard formula of the solution of IVP (1). The operators notation of the IVP (1) is
where \(\texttt {D}= a\partial _{xx} + b\partial _{xy} + c\partial _{yy}\) is differential operator; \(\tilde{\texttt {E}} u = u(0,y), \tilde{\texttt {E}} \partial _x u = \left( \frac{\partial u}{\partial x} \right) _{x=0}\) are initial evaluation operators; and \(\alpha _1, \alpha _2\) are initial data of the given IVP. One can factor the second order differential operator into linear operator [13] and then the given IVP for PDE (2) can be expressed as follows
Now we present main theorem of algorithm for computing the Green?s function of IVP (2) over integrodifferential algebras.
Theorem 1
Let \((\mathcal {F},\texttt {D},\texttt {A})\) be an integrodifferential algebra. Suppose \(\texttt {D}= a\partial _{xx} + b\partial _{xy} + c\partial _{yy}\) is a linear partial differential operator of second order. Then the IVP (2) has the unique solution as follows, for \(m_1 \ne m_2\),
for \(m_1 = m_2\),
here \(m_1 = \frac{b}{2a} + \frac{\sqrt{b^24ac}}{2a}, m_2 = \frac{b}{2a}  \frac{\sqrt{b^24ac}}{2a}\), and \(a \ne 0,b,c \in \mathbb {R}\).
Proof
See [13, Theorem 5].\(\square\)
Main text
This section discusses about the Maple implementation of IVPs for PDEs, IVPforPDE package and its pseudocode. In this implementation, various data types are created to compute the Green’s function, namely PartialDiffOperator(a1,a2,a3), where a1,a2,a3 are the coefficients of the differential operator as given in Eq. (1); and GreensFunction(ParDiffOp), where ParDiffOp is the partial differential operator as given Eq. (2). Now finally, we have ExactSolution(ParDiffOp, forcefun, alpha, beta), where forcefun is the forcing function, alpha, beta are the initial data. We also created ApplyPartialDiffOp(ParDiffOp,sol) to verify the solution sol. This Maple package is available with example worksheet at http://www.sinivasaraothota.webs.com/research.
PseudoCode
The following pseudocode gives the exact solution of a given IVP for PDEs of second order.
The following pseudocode shows how to compute the Green?s function of a given IVP for PDEs of second order of the type in Eq. (1).
Maple implementation
The following procedure gives partial differential operator of the given IVP.
The following procedure produces the Green’s function
Using the following data type, one can obtain the exact solution of the given IVP.
One can also verify the solution using the following procedure.
Results
In this section, we show sample computations using the implementation of IVP. Recall couple of examples from [13] to verify the package. However, there are several example available in [32,33,34] to verify the Maple implementation.
Example 1
Consider second order IVP of the form [13] to demonstrate the Maple implementation.
From Eq. (3), we have \(a_1=1, a_2=1, a_3=6, f(x,y)=y \cos x, \alpha =y1, \beta = y^2\) Using Maple,
> PDO := PartialDiffOperator(1, 1, 6);
> G := GreensFunction(PDO)
> Sol := ExactSolution(PDO, y*cos(x), y1, y\(\wedge\)2);
> ApplyPartialDiffOp(PDO, G);
> ApplyPartialDiffOp(PDO, Sol);
Example 2
Consider second order IVP of the form [13]
The exact solution of IVP (4) as
Using the Maple implementation, we have:
> PDO:=ParialDiffOperator(4,4,1);
> g:=GreensFunction(PDO);
> u:=ExactSolution(PDO,16*log(x+2*y),0,0);
> ApplyPartialDiffOp(PDO,u);
> ApplyPartialDiffOp(PDOp,g);
Limitations
The algorithm in [13] is focused on the IVP for regular linear PDEs, hence the implemented maple package, IVPforPDE, presented in this paper is valid for the regular linear PDEs with initial conditions. We have also presented a syntax to check the validity of solution of a given problem.
Availability of data and materials
The datasets generated and analyzed during the current study are presented in this manuscript.
Abbreviations
 IVP:

Initial value problem
 PDE:

Partial differential equation
References
Thota S, Kumar SD. Solving system of higherorder linear differential equations on the level of operators. Int J Pure Appl Math. 2016;106(1):11–21.
Thota S, Kumar SD. On a mixed interpolation with integral conditions at arbitrary nodes. Cogent Math. 2016;3(1):1151613.
Thota S, Kumar SD. Symbolic algorithm for a system of differentialalgebraic equations. Kyungpook Math J. 2016;56(4):1141–60.
Thota S, Kumar SD. Maple implementation of symbolic methods for initial value problems. In: Research for resurgence, vol. 1, no. 1; 2018. pp. 21–39. ISBN: 9789383861125.
Thota S. On a new symbolic method for initial value problems for systems of higherorder linear differential equations. Int J Math Models Methods Appl Sci. 2018;12:194–202.
Thota S. A symbolic algorithm for polynomial interpolation with integral conditions. Appl Math Inform Sci. 2018;12(5):995–1000.
Thota S. Initial value problems for system of differentialalgebraic equations in Maple. BMC Res Notes. 2018;11:651.
Thota S. On a symbolic method for error estimation of a mixed interpolation. Kyungpook Math J. 2018;58(3):453–62.
Thota S. On a symbolic method for fully inhomogeneous boundary value problems. Kyungpook Math J. 2019;59(1):13–22.
Thota S. A symbolic algorithm for polynomial interpolation with stieltjes conditions in maple. Proc Inst Appl Math. 2019;8(2):112–20.
Thota S. On a new symbolic method for solving twopoint boundary value problems with variable coefficients. Int J Math Comput Simul. 2019;13:160–4.
Thota S, Shanmugasundaram P. On a symbolic method for neutral functionaldifferential equations with proportional delays. Cogent Math Stat. 2020;7(1):1813961.
Thota S, Kumar SD. Symbolic algorithm to solve initial value problems for partial differential equations. Bull Comput Appl Math. 2020;8(1):1–24.
Thota S. On solving system of linear differentialalgebraic equations using reduction algorithm. Int J Math Math Sci. 2020;10. ArticleID 6671926.
Thota S, Kumar SD. A new reduction algorithm for differentialalgebraic systems with power series coefficients. Inform Sci Lett. 2021;10(1):59–66.
Thota S. maple implementation of a symbolic method for fully inhomogeneous boundary value problems. Int J Appl Math Comput Sci Syst Eng. 2021;3:1–5.
Thota S. Maple implementation for reducing differentialalgebraic systems, conference on mathematics for the advancement of science, technology and education, March 5–6, at Ethiopian Mathematics Professionals Association (EMPA). Department of Mathematics: College of Natural and Computational Sciences, Addis Ababa University, Ethiopia; 2021.
Thota S. Maple implementation of a symbolic method for fully inhomogeneous boundary value problems. 2021 International Conference on Mathematics and Computers in Science and Engineering (MACISE 2021), January 18–20, 2021, Madrid, Spain.
Thota S, Kumar SD. Maple implementation of a reduction algorithm for differentialalgebraic systems with variable coefficients. In: International Conference on Recent Trends in Engineering & Technology and Quest for Sustainable Development, May 13–14. Institute of Technology. Ethiopia: Ambo University; 2019.
Thota S. A new symbolic method for solving twopoint boundary value problems, The Hawassa Math&Stat Conference 2019 (February), pp. 11–15, . Hawassa University. Ethiopia: Hawassa; 2019.
Thota S, Kumar SD. Maple Package of Initial Value Problem for System of DifferentialAlgebraic Equations. National Seminar on Applications of Scientific and Statistical Software in Research, 30–31 March. The School of Sciences. Allahabad, India: Uttar Pradesh Rajarshi Tandon Open University; 2017.
Thota S, Kumar SD. A new method for general solution of system of higherorder linear differential equations. International Conference on Inter Disciplinary Research in Engineering and Technology, April 29–30, 2015, New Delhi, India.
Thota S, Kumar SD. Symbolic method for polynomial interpolation with Stieltjes conditions. International Conference on Frontiers in Mathematics, March 26–28. Guwahati University. India: Guwahati; 2015.
Thota S, Kumar SD. A new reduction algorithm for systems of linear differentialalgebraic equations. In: National Conference on Applications of Mathematics in Engineering and Sciences, November 29–30, . Motilal Nehru National Institute of Technology. India: Allahabad; 2014.
Thota S, Rosenkranz M, Kumar SD. Solving systems of linear differential equations over integrodifferential algebras. In: International Conference on Applications of Computer Algebra, June 24–29, 2012, Sofia, Bulgaria.
Thota S, Kumar SD. On a new symbolic method for solving linear boundary problems via noncommutative Groebner bases. International Conference on Recent Advances in Electronics & Computer Engineering, December 17–18. Eternal University. India: Himachal Pradesh; 2011.
Aleksandar D. Analytical solutions of PDEs using PDE tools in Maple (2007).
Falade KI. Introduction to computational algorithm, Numerical and computational research laboratory KUST. 2021. https://doi.org/10.13140/RG.2.2.36674.09920.
George A. Articolo: partial differential equations and boundary value problems with maple. 2nd Edn. 2009. Academic Press is an imprint of Elsevier.
Monagan MB, Geddes KO, Heal KM, Labahn G, Vorkoetter SM, McCarron J, DeMarco P. Maple 9 Advanced Programming Guide. 2003.
Mark S. Gockenbach. Maple tutorial on partial differential equations: analytical and numerical methods (SIAM, 2010), 2nd edition.
Jain RK, Iyengar SRK. Advanced engineering mathematics. 3rd ed. India: Narosa Publishing House; 2008.
Grigoryan V. Partial differential equations. Department of Mathematics: University of California, USA; 2010.
Pinchover Y, Rubinstein J. An introduction to partial differential equations. Cambridge: Cambridge University Press; 2015.
Acknowledgements
The author is thankful to the reviewer and editor for providing valuable inputs to improve the quality and present format of manuscript.
Authors’ information
Dr. Srinivasarao Thota completed his M.Sc. in Mathematics from Indian Institute of Technology (IIT) Madras, India and Ph.D. in Mathematics from Motilal Nehru National institute of Technology (NIT) Allahabad, India. Srinivasarao Thota’s area of research interests are Computer Algebra (symbolic methods for differential equations), Numerical Analysis (root finding algorithms), Mathematical Modeling (ecology). He has published more than 40 research papers in various international journals and presented his research work at several international conferences as oral presenter and invited/keynote/guest speaker in different countries. Presently working at department of applied mathematics, Adama Science and Technology University, Ethiopia.
Funding
Not applicable.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Ethics approval and consent to participate
Not applicable.
Consent for publications
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.
About this article
Cite this article
Thota, S. On solving initial value problems for partial differential equations in maple. BMC Res Notes 14, 307 (2021). https://doi.org/10.1186/s13104021057154
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13104021057154
Keywords
 Initial value problems
 Maple implementation
 Symbolic algorithm
 Partial differential equations
JEL Classification
 35F10
 35G10
 15A29