4/5/2023 0 Comments Linear function equation makerThe integrand involves the operators of heat propagation S ^ and translation Θ ^, which act on the function f ( x ). In particular, we derived the integral form of the particular solution F ( x ) = ( β 2 − ( ∂ x + α ) 2 ) − ν f ( x ) for real values of ν. We have obtained solutions for some ordinary DE of non-integer order with shifted derivatives. Operational Approach and Orthogonal Polynomials Eventually, we will provide the results and the conclusions. In every section we will consider examples of solutions with various initial functions, such as the functions f ( x ) = x n, f ( x ) = ∑ n c n x n, f ( x ) = e − x 2, f ( x ) = e − γ x, f ( x ) = ∑ k x k e γ x, f ( x ) = W 0 ( − x 2, 2 ). In the fourth section we will consider the operational solutions for some PDE in particular, we will consider the evolution partial differential equations of Schrödinger and Black–Scholes types. In the third section we will construct convolution forms of solutions for DE with the help of special functions and integral transforms. In the second section we will apply the orthogonal polynomials and inverse differential operators to find the solution of some non-integer order DE. In the first section we will explore generalized Hermite and Laguerre polynomials, the inverse derivative operator, the Laguerre derivative, and the relations between them we will also touch on the Appèl polynomials. The structure of the manuscript is as follows. The obtained solutions were formulated in terms of series of generalized forms of orthogonal polynomials of Hermite, Laguerre, more general Appèl, and some other polynomials, special functions of hyperbolic, elliptic Weierstrass and Jacobi-type, cylindrical Bessel-type, and generalized Airy-type functions. The operational exponent is also applied when describing the fundamentals of structures in nature, including elementary particles and quarks such modern mathematical instruments are also used for the theoretical study of neutrino mixing and for analysis of relevant experimental data. In the context of the operational approach, the operational definitions for the polynomials through the operational exponent are very useful. The method of operational solution of DE demonstrated in is applicable to a wide spectrum of physical problems, described by linear partial differential equations (PDE), such as propagation and radiation from charged particles, heat diffusion, including processes not described by Fourier law, and others. Examples of the solution of DE of non-integer order and of PDE are considered with various initial functions, such as polynomial, exponential, and their combinations. The Schrödinger and the Black–Scholes-like evolution equations and solved with the help of the operational technique. Some linear partial differential equations (PDE) are also explored by the operational method. Special functions are employed to write solutions of DE in convolution form. Operational definitions of these polynomials are used in the context of the operational approach. The generalized forms of Laguerre and Hermite orthogonal polynomials as members of more general Appèl polynomial family are used to find the solutions. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms and the exponent operators. A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed.
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