Structuring Constitutive Equations of few common Physical Laws Vis-à-vis Types of Impulse Response Functions /Memory Kernels | Chapter 2 | Current Perspective to Physical Science Research Vol. 1

 We will study the composition of constituent expressions of various tangible laws accompanying impulse response function or thought kernel h(t) as plain bordering a river function, singular power-regulation decay function, non-singular capacity law decay function, Mittag-Leffler function, pure epidemic function and stretched epidemic function. The motivation to have this chapter search out discuss, issues about using unique and non-singular functions as fundamental impulse response function or thought kernels; in basic progress equation in several process action-and its suggestions to obtain constituent equations for differing systems. This gives a inference of system studies. We will restrict our reasoning to simple constitutive equatings of few common tangible laws that we deal in common studies. We will study two types of system accompanying evolution equation delineated as convolution that is y(t) = h(t)*x(t) (from Causality Principle). The physical laws that we argue with miscellaneous types of memory kernel are capacitor or dielectric entertainment/force/stress strain rate equations, population growth/atomic reaction equations, spread equations and wave equations. First seeing the cause (input) x(t) is equivalent to rate of change of some other physical load, i.e. x(t) ∝ f(1) (t) and second is a plan where output reaction i.e. y(t) is proportional to rate of change of cause (recommendation) x(t) i.e. y(t) ∝ x(1) (t). We note the first type of structure is like ‘response current’ to a change in applied physical ability observed in dielectric relaxations and capacitor or force to rate of change of impetus or stress and strain rate equations.  The second one complements to population growth or atomic reaction type system.  The corollary to second type of method we study where cause X is discharged by L [x] ; where L denotes a ‘Linear controller’ and y(t) ∝ x(1) (t). With L as Laplacian operator L ≡ ∇ 2, we see that we will generate various types of spread equations.  Extending further making effect as y(t) ∝ x(2) (t), we will write miscellaneous types of wave-equations. We will obtain formation of these basic essential equations accompanying zero-memory case place the memory-essence or impulse response function is a plain bordering a river-function and memorized relaxation cases accompanying singular and non-unique memory kernels or impulse reaction functions that decays with occasion. These decaying functions used as thought kernel gives a existence in which memory fades as occasion grows. However, the question stands is the memory kernel be of unique or non-singular function? We will see that for the nothing-memory case place the memory kernel is a opening function (singular in type) returns classical constitutive equatings for system that we see and use in every day physics but with the case place memory essence is other than delta function we catch constitutive equations accompanying fractional products and fractional integrations, different from what we see classically. We will note that singular function that we use beyond any doubt-decaying memory seed gives rise to combination to classical constitutive equatings where allure fractional counterpart replaces number-order (classical) derivative or integral movement. We will see that non-unique memory kernel gives be even with more complicated essential equations as weighted limitless series total of repeated integrations or weighted succession sum of partial integrations.

Author(s) Details:

Shantanu Das,
Bhabha Atomic Research Center, Mumbai, India and Department of Physics, Jadavpur University, Kolkata, India.

Please see the link here: https://stm.bookpi.org/CPPSR-V1/article/view/11916