New Algorithms to Solve Integral Equations Automatically

Integral equations come from a wide range of applications. Laplace transform has been playing an important role in mathematics; it is very powerful and widely used in solving integral equations, however, such a traditional method suffers a serious drawback, which is the calculation of inverse Laplace transform. Such a kind of inverse calculation is problematic or impossible, except some very simple functions. Sumudu transform is a new integral transform with nice features like Laplace transform, in addition, it provides new methodology for problem solving. In this work, a new computational method is proposed to solve integral equations, the new method incorporates useful features from both Laplace transform and Sumudu transform such that the calculation of the inverse Laplace transform is avoided. In addition, it is demonstrated with implementations that the new method and techniques presented in this work can be implemented in computer algebra systems such as Maple to solve Volterra convolution integral equations and mixed differential Volterra convolution integral equations automatically

[1]  Bruno Salvy,et al.  GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable , 1994, TOMS.

[2]  Adem Kilicman,et al.  Some Remarks on the Sumudu and Laplace Transforms and Applications to Differential Equations , 2012 .

[3]  Fethi Bin Muhammed Belgacem,et al.  Introducing and Analysing Deeper Sumudu Properties , 2006 .

[4]  Robert A. Ravenscroft.  Rational Generating Function Applications in Maple , 1994 .

[5]  Jun Zhang,et al.  Sumudu Transform for Automatic Mathematical Proof and Identity Generation , 2018 .

[6]  Venkat R. Subramanian,et al.  Computational Methods in Chemical Engineering with Maple , 2010 .

[7]  Brigitte Maier,et al.  An Introduction To Transform Theory , 2016 .

[8]  Jun Zhang,et al.  Incorporating Generating Functions to Computational Science Education , 2016, 2016 International Conference on Computational Science and Computational Intelligence (CSCI).

[9]  A. A. Karaballi,et al.  Sumudu transform fundamental properties investigations and applications. , 2006 .

[10]  R. A. Ravenscroft,et al.  Symbolic summation with generating functions , 1989, ISSAC '89.

[11]  Jun Zhang.  Sumudu transform based coefficients calculation , 2008 .

[12]  Jun Zhang,et al.  New Techniques to Solve Differential Equations Automatically , 2017, 2017 International Conference on Computational Science and Computational Intelligence (CSCI).

[13]  Jun Zhang,et al.  A Sumudu based algorithm for solving differential equations , 2007, Comput. Sci. J. Moldova.

[14]  Charles L. Epstein,et al.  The Bad Truth about Laplace's Transform , 2008, SIAM Rev..