Analytical and Numerical Solutions of Certain Differential Equations of Integer and Fractional Order

Abstract

In this thesis, we have solved the differential equations of integral and fractional newlineorders. In chapter 1, we have given brief introduction about the origin of fractional newlinecalculus, motivation, objectives, research gap, and layout of the work. In chapter 2, we newlinehave explained the three methodologies Lie symmetry analysis, homotopy analysis newlinemethod, and reduced differential transform method. In chapter 3, 4, 5, we have applied newlinethe Lie symmetry analysis to Pochhammer-Chree (PC) Equation, Time-Fractional newlineModified Equal Width Wave (TFMEWW) Equation, Kudryashov-Sinelshchilov (KS) newlineEquation, Conduction-Dispersion (CD) Equation, and Time-Fractional Evolution newlineSystem. Explicit power series solution has been derived for PC equation, TFMEWW newlineequation, KS equation, CD equation, and Time-Fractional evolution system and newlineinvariant solution are derived for time-fractional PC equation. Then, conservation laws newlinehave been discussed for KS equation, CD equation, and time-fractional Evolution newlineSystem. In chapter 6, homotopy analysis has been applied to Nonlinear Schrodinger newlineEquation, and Time-Fractional Zakharov Plasma System. Periodic and solitary wave newlinesolution has been derived for nonlinear Schrodinger equation and found to be in good newlineagreement with the classical system. Also approximate analytic solution for Time- newlineFractional Zakharov Plasma System has been studied by HAM .In chapter 7, we have newlineapplied reduced differential transform method to time-fractional coupled systems and newlineobtained solutions are found to be exact when compared with the classical model of the newlineconcerned fractional differential system. After that conclusion and future scope is newlinediscussed in the last section. newline newline