Error and Complexity in Numerical Methods

Lakshmikantham, V.; Sen, Sagar

doi:10.1016/s0076-5392(05)80056-6

Cited by 4 publications

(17 citation statements)

References 3 publications

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“…Besides, quality of a numerical solution -a must for any engineering use as an engineer needs a solution in numbers not in mathematical symbols -implying computational relative error bounds needs to be determined. This quality (the error bounds) allows us to compare relative quality of 2 or more different algorithms -both fractional and corresponding integer order [2].…”

Section: Introductionmentioning

confidence: 99%

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Can Fractional Calculus be Generalized: Problems and Efforts

Sen¹,

Devi

Kumar³

2018

Eur. J. Pure Appl. Math.

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Fractional order calculus always includes integer-order too. The question that crops up is: Can it be a widely accepted generalized version of classical calculus? We attempt to highlight the current problems that come in the way to define the fractional calculus that will be universally accepted as a perfect generalized version of integer-order calculus and to point out the efforts in this direction. Also, we discuss the question: Given a non-integer fractional order differential equation as a mathematical model can we readily write the corresponding physical model and vice versa in the same way as we traditionally do for classical differential equations? We demonstrate numerically computationally the pros and cons while addressing the questions keeping in the background the generalization of the inverse of a matrix.

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Section: Introductionmentioning

confidence: 99%

“…Cost (computational complexity) of the numerical solution also needs to be obtained. This cost permits us to choose that algorithm out of 2 or more algorithms, which is least expensive computationally [2].…”

Section: Introductionmentioning

confidence: 99%

Can Fractional Calculus be Generalized: Problems and Efforts

Sen¹,

Devi

Kumar³

2018

Eur. J. Pure Appl. Math.

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show abstract

“…Statistical randomness does not necessarily mean true randomness [1,2]. Strictly speaking, any number which is the outcome of a process artificial or natural can be at best called a pseudo-random number (pseudo implying false) and can never be a true random number.…”

mentioning

confidence: 99%

“…The pseudo-random numbers are good enough to be used in ant system approach, genetic algorithms, Monte Carlo methods, any evolutionary process or any other randomized algorithm to solve many vitally important computational problems, specifically NP-hard [1,3] problems, such as the TSP, in polynomial-time.…”

mentioning

confidence: 99%