Abstract
Framelets theory has been well studied in many applications in image processing, data recovery and computational analysis due to the key properties of framelets such as sparse representation and accuracy in coefficients recovery in the area of numerical and computational theory. This work is devoted to shedding some light on the benefits of using such framelets in the area of numerical computations of integral equations. We introduce a new numerical method for solving Volterra integral equations. It is based on pseudo-spline quasi-affine tight framelet systems generated via the oblique extension principles. The resulting system is converted into matrix equations via these generators. We present examples of the generated pseudo-splines quasi-affine tight framelet systems. Some numerical results to validate the proposed method are presented to illustrate the efficiency and accuracy of the method.
Highlights
Many natural science problems are modeled by Volterra integral equations, which has brought them much attention from scientists in numerical analysis
The redundancy property of framelets has been used for many applications in science and engineering disciplines, for example, in the analysis of the Gibbs phenomenon and numerical solutions of various types of integral equations, in time–frequency theory for image analysis, multifilter designs in electrical engineering, the theory of nonshift and shift-invariant spaces, and many other areas
To validate the accuracy of our method, we present the following example of Volterra integral equations
Summary
Many natural science problems are modeled by Volterra integral equations, which has brought them much attention from scientists in numerical analysis. Some approximations work better with redundant expansions such as the biorthogonal wavelet (or framelet) expansions. The redundancy property of framelets has been used for many applications in science and engineering disciplines, for example, in the analysis of the Gibbs phenomenon and numerical solutions of various types of integral equations (see, e.g., [1,2,3,4,5,6]), in time–frequency theory for image analysis, multifilter designs in electrical engineering, the theory of nonshift and shift-invariant spaces, and many other areas We have more freedom in building efficient and accurate recovery
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
