Existence of positive solutions for a class of BVPs in Banach spaces

In this work, we use index xed point theory for perturbation of expan- sive mappings by l-set contractions to study the existence of bounded positive solutions for a class of two-point boundary value problem (BVP) associated to second-order nonlinear di erential equation on the positive half-line. The nonlin- earity, which may exhibit a singularity at the origin, is written as a sum of two functions which behave di erently. These functions, depend on the solution and its derivative, take values in a general Banach space and have at most polynomial growth. An example to illustrate the main results is given.

A New Framework for the Determination of the Eigenvalues and Eigenfunctions of the Quantum Harmonic Oscillator

This study was designed to obtain the energy eigenvalues and the corresponding Eigenfunctions of the Quantum Harmonic oscillator through an alternative approach. Starting with an appropriate family of solutions to a relevant linear di erential equation, we recover the Schr¨odinger Equation together with its eigenvalues and eigenfunctions of the Quantum Harmonic Oscillator via the use of Gram Schmidt orthogonalization process in the usual Hilbert space. Significantly, it was found that there exists two separate sequences arising from the Gram Schmidt Orthogonalization process; one in respect of the even eigenfunctions and the other in respect of the odd eigenfunctions.

New soliton solutions of nonlinear Kudryashov's equation via Improved tan 􀀀 ( ) 2 -expansion approach in optical ber

In this article, a newly introduced model nonlinear Kudryashov's equation with anti-cubic non- linearity is considered for extraction of soliton solutions. This model is utilized to depict the propagation of modulated envelope signals which disseminate with some group velocity. To nd a solution, an appropriate traveling wave hypothesis is used to covert the given model into a nonlinear ordinary di erential equation. An analytical technique, the Improved tan ( ) 2 - expansion approach has been employed on the governing model to construct many new forms of dark soliton, singular soliton, periodic soliton, dark-singular combo soliton and rational solu- tion. The constraint conditions for the existence of these solitons have also been provided. The physical signi cance of the proposed equation has been provided with a graphical representation of the constructed solutions.

A Study of SDEs Driven by Brownian Motion and Fractional Brownian Motion

In this thesis, we mainly study some properties for certain stochastic di↵er-ential equations.The types of stochastic di↵erential equations we are interested in are (i) stochastic di↵erential equations driven by Brownian motion, (ii) stochastic functional di↵erential equations driven by fractional Brownian motion, (iii) McKean-Vlasov stochastic di↵erential equations driven by Brownian motion,(iv) McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion.The properties we investigate include the weak approximation rate of Euler-Maruyama scheme, the central limit theorem and moderate deviation principle for McKean-Vlasov stochastic di↵erential equations. Additionally, we investigate the existence and uniqueness of solution to McKean-Vlasov stochastic di↵erential equations driven by fractional Brownian motion, and then the Bismut formula of Lion’s derivatives for this model is also obtained.The crucial method we utilised to establish the weak approximation rate of Euler-Maruyama scheme for stochastic equations with irregular drift is the Girsanov transformation. More precisely, giving a reference stochastic equa-tions, we construct the equivalent expressions between the aim stochastic equations and associated numerical stochastic equations in another proba-bility spaces in view of the Girsanov theorem.For the Mckean-Vlasov stochastic di↵erential equation model, we first construct the moderate deviation principle for the law of the approxima-tion stochastic di↵erential equation in view of the weak convergence method. Subsequently, we show that the approximation stochastic equations and the McKean-Vlasov stochastic di↵erential equations are in the same exponen-tially equivalent family, and then we establish the moderate deviation prin-ciple for this model.Based on the result of Well-posedness for Mckean-Vlasov stochastic di↵er-ential equation driven by fractional Brownian motion, by using the Malliavin analysis, we first establish a general result of the Bismut type formula for Lions derivative, and then we apply this result to the non-degenerate case of this model.

ADDITIONAL SOUGHT-FOR FUNCTIONS AND LOCAL COORDINATE SYSTEMS APPLIED IN THE HEAT CONDUCTIVITY PROBLEMS FOR MULTILAYERED BUILDING STRUCTURES

The solution problems of the additional the sought-for function and additional boundary conditions based when using local coordinate systems, an approximate analytical solution of the heat conduction problem for a double-layer plate is obtained for symmetric boundary conditions of the fi rst kind. The use of the additional sought-for function in the integral method of heat balance makes it possible to reduce the solution of the partial diff erential equation to the integration of an ordinary diff erential equation.

An Intriguing Approach to The Fractional Mellin Transform Method

We propose an adapted Mellin transform method that gives the solution of a fractional di ¤erential equation with variable coefficients in ordinary domain. After we mention a transformation of cosmic time to individual time (CTIT), we explain how it can reduce the problem from fractional form to ordinary form when it is used with Mellin transformation, via an example for 0 < alpha < 1; where alpha is the order of fractional derivative. Then, we give an application of the results.

DYNAMIC CALCULATION OF PRISMATIC SYSTEMS TAKING INTO ACCOUNT INTERNAL FRICTION

The paper shows a possibility of including a frequency-independent viscoelastic design resistance in calculation models representing composite structures with distributed parameters. Diff erential equations of the system motion include parameters of complex rigidity. The research puts forward a solution which helps solve stationary and non-stationary dynamic problems taking internal friction into account. This solution is found by using the calculation algorithm for prismatic shells involving the use of topological matrices. In the process of separation of variables, this approach yields a diff erential equation of linear oscillator motion with frequency-independent viscoelastic resistance.