In the last century, there has been extensive research on our brain and many mathematical models and theories have been developed which describe the dynamical behavior of neurons. One of them is the widely known, Hodgkin-Huxley model. The Hodgkin-Huxley model for space clamp situation (uniform voltage over a patch of nerve membrane) is a mathematical model consisting of 4 nonlinear ordinary differential equations that describe membrane action potentials in neurons. Before this work, these equations could only be solved by numerical techniques and analytical solutions were not found. In this work, efforts are put to find the analytic solution of the Hodgkin-Huxley model by using Homotopy Perturbation Method. Homotopy Perturbation Method was developed by Ji-Huan He (1998) by merging two techniques, the standard homotopy and the perturbation technique for solving linear, nonlinear, initial and boundary value problems. Further, the solution is compared with the experimental results found by Hodgkin and Huxley. In this work, the first-order approximate analytic solution of the space-clamped Hodgkin Huxley model has been computed and algorithm for a higher-order solution is given. For plotting the solution, MATHEMATICA is used. It is found that the first-order solution can describe many key properties of the Hodgkin-Huxley model. Further, besides some differences, the general agreement of the first-order solution of space-clamped Hodgkin-Huxley Equations by Homotopy Perturbation Method with experimental results is good. Homotopy Perturbation The method is proved to be a convenient and efficient method to find an approximate or exact analytic solution of nonlinear differential equations
اجوکی کافی دا وَڈا شاعر: افضال حسین گیلانی Kafi is a genre of Poetry that is found in the poetry of Sufis. It contains accessories such as Naat, Manqabat, and Qasida where the state of distance and the pleasure of connection are maintained. Kafi began with Shah Hussain and many Sufi poets experimented with it. One of them is Syed Afzal Hussain Gilani who has described Kafi as a classical tradition as well as a contemporary one. This article discusses Syed Afzal Hussain Gilani's Kafies.
This study is dedicated to the subject of constructing exact solutions of nonlinear partial differential equations arising in engineering science, mathematical physics, fluid mechanics, quantum mechanics etc. The significance of nonlinear partial differential equations lies in the fact that the mathematical modeling of many real life phenomena involves differential equations. Several new nonlinear evolution equations are reported to appear in recent scientific studies. The investigation of exact solutions of nonlinear partial differential equations is of great value in understanding widely different physical phenomena. In late years, many fractional order evolution equations are applied successfully to model various physical phenomena. The memory effects of the fractional derivatives play an important role in understanding mechanical and electrical attributes of actual materials. The aim of this dissertation is to extract the exact solutions of some important nonlinear partial differential equations of integer order as well as non-integer order, for instance the nonlinear short-pulse equation, nonlinear fractional biological population model, the nonlinear complex Ginzburg-Landau equation, the generalized fractional Zakharov-Kuznetsov equation, time-fractional Chan-Allen equation, the fractional dispersive modified Benjamin-Bona-Mahony equation and (2 + 1)−dimensional solution equation. Some well known integration techniques are implemented in carrying out the travelling wave solutions of these equations. The obtained solutions may be worthwhile for explanation of some physical phenomena accurately.