روایتی بینکاری نظام کے ارتقائی مراحل کا تحقیقی جائزہ

The concept of keeping wealth in a safe place dates to centuries. Ancient civilizations had diverse means of storing wealth in the form of crops, cattle, precious metals etc. The evolution of modern banking practice began with the introduction of receipts which were exchanged against precious metals and coins deposited to goldsmiths for safe keeping. Whenever the need for payments and transactions arose the holder of the receipts used to utilize the receipts as guarantee. The society used to honor these receipts as they carried the same weight as other precious metals. Receipts were swapped in place of precious metals and thus for all practical purposes paper was introduced as currency in the society. With the advent of currency notes the system of traditional banking came into being. Since then the banking system has gone through continuous change. The present banking system is geared up to meet the present and the future requirements of modern age. In the contemporary world money is now being steadily replaced by banknotes, cheques, pay orders, bank draft, ATM cards, debit cards, credit cards, e-banking.

Some Wavelet Schemes for Nonlinear Differential Equations

This work is devoted to the study of wavelet schemes for solving nonlinear differential equations. Most of the scientific and engineering phenomena can be represented in the form of nonlinear differential equations. Over the past few decades, nonlinear differential equations have been the core of research for many researchers and scientists. Owing to the non-availability of exact solutions in many nonlinear physical problems representing complex phenomena, various analytical and non-analytical schemes have been evolved. One of the most recent families of schemes developed for finding solutions of differential equations is Wavelet schemes. These newly revolutionized schemes have few shortcomings, while dealing with nonlinear differential equations. The existing wavelets schemes are being modified and enhanced in this study to overcome these shortcomings. In this study, techniques such as Picard’s Iteration Method, Quasilinearization Method and Method of Steps have been merged with different wavelet schemes, which proved to be very proficient, reliable and effective in handling a large number of nonlinear mathematical problems representing nonlinear differential equations and their systems. These wavelet schemes have been extended for fractional nonlinear differential equations also.