q-linearty and q-linear differantial equations

In science and engineering, mathematical formulas or mathematical models are developed to help explain many events. These models emerge as an equation containing an unknown function and some derivatives of this function. Such equations are called differential equations. Differential equations are a basic branch of analysis that is widely used in mathematics, natural sciences, medicine and engineering. Differential equations are a basic branch of analysis that is widely used in mathematics, natural sciences, medicine and engineering. In addition today, solutions of differential equations and differential equation systems can be found in a computer environment with the help of different programs. The aim here is to make a comparison between the analytical solution and the solution in the computer environment. Linear differential equations, one of the most important sub-branches of differential equations, have a very important share in the application and solution of differential equations. In addition, the properties of the solutions, the definitions and theorems required to obtain the solutions play an important role. In finding the solutions of linear differential equations, a systematic order solution should be sought first. In the given linear differential equation, firstly it is observed whether it is homogeneous or not. Afterwards, if the differential equation is homogeneous, the homogeneous part solution is found. At this stage, the coefficients of the differential equation guide us. Here, the situation of the differential equation being constant coefficient and variable coefficient is taken into consideration. If the specified equation is a first order linear differential equation, solution methods are applied. Separable equations, complete differential equations, integral multiplier method are some of these methods. The solution of linear differential equations of second order or higher order provides us with great convenience when examined on the basis of coefficients. When the given equation is constant coefficient and homogeneous, it can be found by converting it to the characteristic form, which is a practical solution method. However, when there are variable coefficients, this situation may not provide us with the same convenience. That is, these equations cannot be solved by algebraic methods, except for a few special equation classes, and their solutions cannot be expressed in terms of elementary functions. A solution is sought with the power series method, which is a general method for solving these equations. This method is given for second order homogeneous linear differential equations. In this study, the notation similar to quantum analysis will be examined up to this point. Quantum calculus is generally known as derivative without limit notation. The history of quantum calculus dates back to the 1400s, when Leonhard Euler began his studies. Quantum calculus has two main branches: q-like and h-like. This study will also deal with the q-similar expression. The formulas obtained in quantum calculus emerged in the 18th century. The beginning of q-analysis in the modern sense can be considered as the q-Jackson integral published by Jackson in 1910. Afterwards, the integral representations of q-gamma and q-beta functions were discussed by Victor Kac and Pokmen Cheung. Nowadays, special functions of applied mathematics are a new application area of q-calculus. Especially Sturm-Liouville problems are seen to be a subject with many applications in this field. It has been shown that some basic expressions in analysis such as differential, derivative, integral are q-differential, q-derivative, q-integral in quantum calculus. Similarly, just as the antiderivative is obtained from the derivative definition in classical analysis, the q-antiderivative is obtained from the q-derivative and the q-definite integral is obtained from the q-integral in quantum analysis. In addition, it has been seen that the q-like expression of the exponential function, which is frequently used in this study, also provides practical solutions in quantum calculus. In this thesis, the linearity condition of linear differential equations, which is a sub-branch of classical differential equations, is examined within the context of a new paradigm, q-Analysis. First of all, this study consists of five sections. In the first section, the history of analysis, differential equations and q-analysis required for this thesis is mentioned, how it has progressed from the past to the present and how bridges have been established between them are discussed. In the second section, the necessary definitions and theorems for analysis and q-analysis are given. Then, in the third section, q-linearity conditions were determined; The most general equation of q-linear differential equations of nth order was expressed as; a_0 D_q^n y+a_1 D_q^(n-1) y+a_n y=Q Here, a_0,a_1,a_2,…,a_n functions represent the coefficients of the equation. The general expressions of the equation mentioned above, first of order and Q(x)=0, namely, the homogeneous q-linear differential equation with constant coefficients and variable coefficients, are mentioned. In addition, solution methods were evaluated, and within this framework, it was examined which methods were functional or not. Accordingly, when we consider it based on coefficients; The solution of some types of homogeneous q-linear differential equations with variable coefficients by means of q-exact differential equations has been investigated. Similarly, while the integral multiplier method, which is one of the solution methods of classical linear differential equations, is used as a q-integral multiplier in q-analysis; it has been investigated whether this method has an equivalent in the solution of q-linear differential equations. As the third method, the solution method with power series in differential equations and the solution method with q-power series in q-Analysis were determined. Afterwards, in the case of Q(x)=0, that is, the solution of the non-homogeneous q-linear differential equation was examined with the parameter change method, which is one of the solution methods of classical linear differential equations, namely the q-parameter change method in q-Analysis. In the fourth section, the solution methods of higher order constant coefficient homogeneous q-linear differential equations are investigated. One of these methods is the characteristic polynomial conversion method, which is the solution method of the homogeneous part of the linear differential equations with constant coefficients of higher order from classical differential equations. The other method is the most general nth order q-linear differential equation mentioned above, in case n=2, the solution is sought with the q-power series examined in the previous section. In the fifth section, which is the last section, the conclusion, it is determined which method is more functional as a result of the researches and examinations and suggestions are presented.