In this paper, we propose a numerical method based on the generalized hat functions (GHFs) and improved hat functions (IHFs) to find numerical solutions for stochastic Volterra-Fredholm integral equation. To do so, all known and unknown functions are expanded in terms of basic functions and replaced in the original equation. The operational matrices of both basic functions are calculated and em...

Our goal in this section is to define and develop the six basic trigonometric functions. We start with two of the more important transcendental functions, namely the sine function and the cosine function. These are universally denoted by y = sinx and y = cos x respectively. In order to define sinx and cosx, we first need to agree on how we are going to measure angles. An angle, θ, consists of t...

We use the following convention: x, t, t1, t2, r, a, b are real numbers and F , G are partial functions from R to R. Let F be a partial function from R to R and let t be a real number. We say that t is a period of F if and only if: (Def. 1) t 6= 0 and for every x holds x ∈ domF iff x+t ∈ domF and if x ∈ domF, then F (x) = F (x+ t). Let F be a partial function from R to R. We say that F is perio...

The notion of function is indispensable to our understanding of distinctions such as that between being broken and being in working order (for artifacts) and between being diseased and being healthy (for organisms). A clear account of the ontology of functions and functioning is thus an important desideratum for any top-level ontology intended for application to domains such as engineering or m...