Structuring Constitutive Equations of few common Physical Laws Vis-à-vis Types of Impulse Response Functions /Memory Kernels

We will study the formation of constituent expressions of various physical laws with impulse response function or memory kernel h(t) as delta function, singular power-law decay function, non-singular power law decay function, Mittag-Leffler function, pure exponential function and stretched exponential function. The motivation to have this chapter is to discuss, issues about using singular and non-singular functions as basic impulse response function or memory kernels; in basic evolution equation in several process dynamics-and its implications to obtain constituent equations for various systems. This gives a generalization of system studies. We will restrict our analysis to simple constitutive equations of few common physical laws that we deal in everyday studies. We will study two types of system with evolution equation defined as convolution i.e. y(t) = h(t)*x(t) (from Causality Principle). The physical laws that we discuss with various types of memory kernel are capacitor or dielectric relaxation/force/stress strain rate equations, population growth/radioactive decay equations, diffusion equations and wave equations. First considering the cause (input) x(t) is proportional to rate of change of some other physical quantity, i.e. x(t)
f(1) (t) and second is a system where output response i.e. y(t) is proportional to rate of change of cause (input) x(t) i.e. y(t)
x(1) (t). We note the first type of system is like ‘response current’ to a change in applied voltage observed in dielectric relaxations and capacitor or force to rate of change of momentum or stress and strain rate equations. The second one corresponds to population growth or radioactive decay type system. The corollary to second type of system we study where cause X is replaced by L [x] ; where L denotes a ‘Linear operator’ and y(t)
x(1) (t). With L as Laplacian operator L

2, we see that we will be getting various types of diffusion equations. Extending further making effect as y(t)
x(2) (t), we will write various types of wave-equations. We will derive formation of these basic constitutive equations with zero-memory case where the memory-kernel or impulse response function is a delta-function and memorized relaxation cases with singular and non-singular memory kernels or impulse response functions that decays with time. These decaying functions used as memory kernel gives a reality in which memory fades as time grows. However, the question arises is the memory kernel be of singular or non-singular function? We will see that for the zero-memory case where the memory kernel is a delta function (singular in nature) returns classical constitutive equations for system that we know and use in every day physics but with the case where memory kernel is other than delta function we get constitutive equations with fractional derivatives and fractional integrations, different from what we know classically. We will note that singular function that we use for time-decaying memory kernel gives rise to conjugation to classical constitutive equations where its fractional counterpart replaces integer-order (classical) derivative or integral operation. We will see that non-singular memory kernel gives rise to more complicated constitutive equations as weighted infinite series sum of repeated integrations or weighted series sum of fractional integrations.