Ahoy there, matey! Set yer sails for a grand voyage into the realm of fractional calculus, where we be explorin’ the mysterious depths of plasticine body deformations. Avast, it may sound complex, but fear not, for I’ll break it down for ye in true pirate fashion!

Picture yerself a lump of plasticine, aye? When ye press or pull on it, it stretches and squishes, changin’ shape. Now, regular calculus be good for describin’ how things change over whole numbers o’ time, but what if we want to know how our plasticine be changin’ over a fraction o’ time? That’s where fractional calculus comes in!

Instead o’ simple derivatives (that be tellin’ us how things change over a whole unit o’ time), fractional calculus uses fractional derivatives to describe how things be changin’ over a fraction o’ time. Arr, this be a mighty useful tool when ye be studyin’ materials like plasticine that can deform in complex ways.

Ye see, the traditional derivatives be all about how things move and change in integer steps, like how far ye be movin’ in one whole second. But in reality, some materials like plasticine be mighty peculiar, changin’ their shape smoothly and gradually over a fraction o’ time. Aye, think about how it stretches and squashes when ye be pressin’ it real slow and gentle-like.

Fractional calculus be helpin’ us capture this smooth behavior. It be dealin’ with fractional orders o’ differentiation and integration. A fractional derivative, which we call a “dervish” (not to be confused with a real dervish, the whirlin’ Sufi dancer), lets us understand how our plasticine be changin’ over a fraction o’ time.

So when ye be modelin’ deformations o’ plasticine bodies, fractional calculus be a mighty fine approach to understandin’ those gradual, fractional changes in shape. It be helpin’ ye create more accurate and realistic models for yer plasticine adventures!

Now, ye be set to sail the seas of fractional calculus and tame the wild plasticine deformations, arr! May the wind be ever in yer favor, me hearty! Yo-ho-ho!

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