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On solving initial value problems for partial differential equations in maple
BMC Research Notes volumeÂ 14, ArticleÂ number:Â 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,Â [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 inÂ [13]. For more understanding and application on Maple software commands on how to obtain numerical solutions and the plots seeÂ [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, seeÂ [13] for more details. InÂ [13], they presented a new symbolic method/algorithm to find the Greenâ€™s function for a given IVP of linear second order PDEs with constant coefficients. In this method, they focused on computing the Greenâ€™s function using the integrodifferential algebrasÂ [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. Â 1.1, we briefly state the symbolic algorithm of IVP for PDEs, Sect. Â 2 presents the Maple implementation of the algorithm with pseudocode and Maple programming and Sect. Â 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, seeÂ [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 ofÂ (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 IVPÂ (1) not only for static values of f(x,Â y), \(\alpha _1(y),\alpha _2(y)\) but give a standard formula of the solution of IVPÂ (1). The operators notation of the IVPÂ (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 operatorÂ [13] and then the given IVP for PDEÂ (2) can be expressed as follows
Now we present main theorem of algorithm for computing the Green?s function of IVPÂ (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 IVPÂ (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
SeeÂ [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 Greenâ€™s function, namely PartialDiffOperator(a1,a2,a3), where a1,a2,a3 are the coefficients of the differential operator as given in Eq.Â (1); and GreensFunction(ParDiffOp), where ParDiffOp is the partial differential operator as given Eq.Â (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. Â (1).
Maple implementation
The following procedure gives partial differential operator of the given IVP.
The following procedure produces the Greenâ€™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 fromÂ [13] to verify the package. However, there are several example available inÂ [32,33,34] to verify the Maple implementation.
Example 1
Consider second order IVP of the formÂ [13] to demonstrate the Maple implementation.
From Eq.Â (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 formÂ [13]
The exact solution of IVPÂ (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 inÂ [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
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Acknowledgements
The author is thankful to the reviewer and editor for providing valuable inputs to improve the quality and present format of manuscript.
Authorsâ€™ 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 Thotaâ€™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.
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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
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DOI: https://doi.org/10.1186/s13104021057154