On a Symbolic Method for Fully Inhomogeneous Boundary Value Problems (original) (raw)
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This study presents a symbolic approach for solving second order boundary value problems with Stieltjes boundary conditions (integral, differential, and generic boundary conditions). The proposed symbolic method computes the Green’s operator and the Green’s function of the provided boundary value problem on the level of operators by applying the algebra of integro-differential operators. The suggested algorithm will aid in implementing manual calculations in mathematical software programs like Mathematica, Matlab, Singular, Scilab, Maple and others.
Maple Implementation of A Symbolic Method for Fully Inhomogeneous Boundary Value Problems
Int. J. of Applied Mathematics, Computational Science and Systems Engineering, 2021
In this paper, we discuss the maple implementation of a symbolic method for solving a boundary value problem with inho-mogeneous Stieltjes boundary conditions over integro-differential algebras. The implementation includes computing the Green's operator and the Green's function of a given boundary value problem. Sample computations are presented to illustrate the Maple implementation.
A New Symbolic Method for Linear Boundary Value Problems Using Groebner Bases
Citeseer
We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the TH∃OREM∀ system; some code fragments and sample computations are included.
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We present a new version of the Maple package IntDiffOp for Symbolic Computation with boundary problems for linear ordinary differential equations. The solution of boundary problems for linear ordinary differential equations is of great practical importance, and there is a vast literature on their analytic treatment . A new symbolic approach was introduced in [10] and subsequently generalized to a differential algebra setting in . For a recent survey and references on our symbolic approach to boundary problems we refer to . The first implementations were coded in Mathematica/TH∃OREM∀, as an external package in [10] for boundary problems with constant coefficients and as an internal functor in for generic integro-differential algebras. In contrast to the stepwise reduction approach of the Mathematica packages, the IntDiffOp package uses normal forms (up to basis expansion) . In [7] we give a detailed description of the functionality for solving and factoring regular boundary problems (i.e. those having a unique solution for every right hand side), similar to the package in . Moreover, we introduce an algorithmic approach for singular boundary problems and generalized Green's operators .
Green's Functions for Stieltjes Boundary Problems
Proceedings of the 2015 ACM on International Symposium on Symbolic and Algebraic Computation, 2015
Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation points. (2) They allow derivatives of arbitrary order. (3) Global terms in the form of definite integrals are allowed. Assuming the Stieltjes boundary problem is regular (a unique solution exists for every forcing function), there are symbolic methods for computing the associated Green's operator. In the classical case of well-posed two-point boundary value problems, it is known how to transform the Green's operator into the so-called Green's function, the representation usually preferred by physicists and engineers. In this paper we extend this transformation to the whole class of Stieltjes boundary problems. It turns out that the extension (1) leads to more case distinction, (2) implies ill-posed problems and hence distributional terms, (3) has apparently no effect on the structure of the Green's function.
On A New Symbolic Method for Solving Two-point Boundary Value Problems with Variable Coefficients
International Journal of Mathematics and Computers in Simulation, 2019
In this paper, we discuss a simple and efficient symbolic method to find the Green’s function of a two-point boundary value problem for linear ordinary differential equations with inhomogeneous Stieltjes boundary conditions. The proposed method is also applicable to find an approximate solution of a two-point boundary value problem for non-linear differential equations. Certain examples are presented to illustrate the proposed method. The method is easy to implement the manual calculations in commercial mathematical softwares, such as Maple, Mathematica, Singular, SCIlab etc. Implementation of the proposed algorithm in Maple is also discussed and sample computations are shown using the Maple implementation.
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Lecture Notes in Computer Science, 2009
We describe a symbolic framework for treating linear boundary problems with a generic implementation in the Theorema system. For ordinary differential equations, the operations implemented include computing Green's operators, composing boundary problems and integrodifferential operators, and factoring boundary problems. Based on our factorization approach, we also present some first steps for symbolically computing Green's operators of simple boundary problems for partial differential equations with constant coefficients. After summarizing the theoretical background on abstract boundary problems, we outline an algebraic structure for partial integro-differential operators. Finally, we describe the implementation in Theorema, which relies on functors for building up the computational domains, and we illustrate it with some sample computations including the unbounded wave equation.
Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases
We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integro-differential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the Theorema system; some code fragments...
Journal of Symbolic Computation, 2008
We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct a ring of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators.
Symbolic Algorithm to Solve Initial Value Problems for Partial Differential Equations
Bulletin of Computational Applied Mathematics, 2020
In this paper, a new symbolic algorithm to find the Green's function of a given initial value problem for linear partial differential equations of second order with constant coefficients is discussed. Same algorithm also works for n-th order partial differential equations. We employ the integro-differential algebra to express the initial value problems and the Green's function. Some examples are presented to illustrate the proposed method compared with other existing method. Implementation of the proposed algorithm in Maple is discussed and sample computations are shown.