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The main interest of this paper is to describe new computational techniques for solving nonlinear problems using g -fractional differential operators. First, we introduce the concept of the g -conformable fractional differential operator on g -semiring. Then, the mean value theorem and Rolle’s theorem for g -conformable fractional differential operator are investigated. Moreover, we consider the exact solution of g -fractional differential equations.

We present a general operator method based on the advanced technique of the inverse derivative operator for solving a wide range of problems described by some classes of differential equations. We construct and use inverse differential operators to solve several differential equations. We obtain operator identities involving an inverse derivative operator, integral transformations, and generalized forms of orthogonal polynomials and special functions. We present examples of using the operator method to construct solutions of equations containing linear and quadratic forms of a pair of operators satisfying Heisenberg-type relations and solutions of various modifications of partial differential equations of the Fourier heat conduction type, Fokker–Planck type, Black–Scholes type, etc. We demonstrate using the operator technique to solve several physical problems related to the charge motion in quantum mechanics, heat propagation, and the dynamics of the beams in accelerators.

In this article we present uniqueness, existence, and L p -estimates of the quasilinear stochastic partial differential equations driven by Levy processes of the type (0.1) d u = ( L u + F ( u ) ) d t + G k ( u ) d Z t k , where L is a pseudo-differential operator and Z k are independent Levy processes ( k = 1 , 2 , ⋯ ) . The operator L is random and may depend also on time and space variables. In particular, our results include an L p -theory of 2 m -order SPDEs with coefficients measurable in ( ω , t ) and continuous in x .

This article devoted to study periodic M-hypoelliptic pseudo-differential operators on torus $$\mathbb {T}^{n}= (\frac{\mathbb {R}}{2\pi \mathbb {Z}})^{n}.$$ More precisely, we consider toroidal M-hypoelliptic symbols and we construct the minimal and maximal extension on $$L^{p}(\mathbb {T}^{n}),$$ $$1 < p

We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator. Our main result shows the existence of infinitely many nodal solutions in the case where we do not impose any restriction on the behavior of the reaction term f at infinity. This result completes some recent papers. The studied differential operators incorporate as special cases the p-Laplacian, the (p, q)-differential operator and the generalized p-mean curvature differential operator. Our approach is based on variational methods combining suitable truncation and comparison techniques.

With the use of similar operator method, in this paper we study spectral properties of fourth order differential operator with periodic and antiperiodic boundary conditions. We obtain asymptotic of a spectrum and estimates of equiconvergence of spectral decompositions for the investigated operator. We also construct a semigroup of operators which has the opposite differential operator as a generator.

The main purpose of this paper is to introduce new generalized differential operator Aμm, λ(α,β)f(z) defined in the open unit disc U = {z ∈ : |z| < 1}. We then, using this operator to introduce novel subclass Ωm*(δ,λ,α,β,b) by using the operator Aμm, λ(α,β)f(z). Then, we discuss coefficient estimates growth and distortion theorems, closure theorems and integral operator.

We consider the formal prolate spheroid differential operator on a finite symmetric interval and describe all its self-adjoint boundary conditions. Only one of these boundary conditions corresponds to a self-adjoint differential operator which commute with the Fourier operator truncated on the considered finite symmetric interval.

In this chapter, existence of solution of some well-known partial differential equations with boundary conditions is studied.

In the present paper we introduce sufficient conditions for subordination and superordination involving the operator DR λ m , n $DR_\lambda ^{m,n} $ and also we obtain sandwich-type results.

In this paper, we establish the $M^k$-type sharp maximal function inequalities for the Toeplitz type operator associated to the pseudo-differential operator. As an application, we obtain the boundedness of the operator on Lebesgue and Morrey spaces.

Substantive foundations are offered for solution of differential equations with fixed elements of coefficient system and continuous measurable functions of the right part.

In this report we derive moments of several discrete distributions via a differential operator. The first part of the report is devoted to the moments of truncated classical distributions. In the second part we present the moments for inflated and deformed power series distributions. Finally we use a differential operator to derive moments of two modified negative binomial distributions.

Estimates in various Lebesgue spaces \(L_s(G)\), \(1\le s\le \infty \), are obtained for the root functions of an operator which relates to the differential operation \(-u''+p(x)u'+q(x)u\), \(x\in G=(a,b)\), with complex-valued singular coefficients. Among these estimates there are also the so-called anti-a priori estimates that link the root functions in the same chain. It is supposed that p(x) and q(x) belong locally to the spaces \(L_2\) and \(W_2^{-1}\), respectively, may have singularities at the end-points of G, and \(q(x)=q_1(x)+Q'(x)\) while \(Q(x), p(x), Q^2(x)w(x), p^2(x)w(x), q_1(x)w(x)\) are integrable on the whole interval G with \(w(x)=(x-a)(b-x)\).

We develop the local inequalities with new weights for the potential operator applied to differential forms. We also prove the global weighted norm inequalities for the potential operator in averaging domains and explore applications of our new results.

The Null Space Pursuit (NSP) algorithm based on the differential operator is an important method of signal denoising and signal separation. In this paper, we propose an arbitrary order differential operator and use it in a complex signal which is a sum of simple signal. By solving an optimization problem, we also estimate the parameters of the differential operator. Finally, we confirmed the practicability of the algorithm through experimental simulation.

We prove a theorem on the smoothness of generalized solutions of differential equations with operator coefficients.

" PDF File" DOI : http://dx.doi.org/10.22342/jims.13.1.72.27-38

In this paper a new method of construction of a solution for a wide class of fractional order differential equations is presented. This method is a generalization of the method of operator algorithms to fractional order differential equations. Along with this, new types of special functions are introduced and some of their properties are studied.

In this article, fractional exponential operator is considered as a general approach for solving partial fractional differential equations. An integral representation for this operator is derived from the Bromwich integral for the inverse Mellin transform. Also, effectiveness of this operator for obtaining the formal solution of system of diffusion equations is discussed. Copyright © 2011 John Wiley & Sons, Ltd.

A linear differential/integral operator is associated to a rational matrix in a natural way, and the behavior is defined to be the kernel of this operator. Conditions under which two rational matrices have the same behavior are derived.

In the class of holomorphic vector functions, we determine the conditions of solvability of the boundaryvalue problem for a class of second-order operator differential equations expressed in terms of the operator coefficients appearing both in the equation and in the boundary condition.

We construct local ℒ-splines with uniform knots that preserve subsets from the kernel of a linear differential operator ℒ of order r with constant real coefficients and pairwise distinct roots of the characteristic polynomial. Pointwise estimates are found for the error of approximation by the constructed ℒ-splines on classes of functions defined by differential operators of orders smaller than r.

The necessary and sufficient conditions for the existence of the variational multiplier for the operator of one system of nonlinear evolutionary partial differential equations have been obtained and the algorithm for its construction is proposed.

The oblique derivative problem is addressed for an elliptic operator differential equation with variable coefficients in a smooth domain. Several conditions are obtained, guaranteing the maximal regularity, the Fredholm property, and the positivity of this problem in vector-valued Lp-spaces. The principal part of the corresponding differential operator is nonselfadjoint. We show the discreteness of the spectrum and completeness of the root elements of this differential operator. These results are applied to anisotropic elliptic equations.

In this paper, we use q-dierential operator to recover the finite Heine 2'1 transformations given in (3). Applying that, we also obtain some terminating q-series transformation formulas.

In this paper, the solutions of initial value problems for a class of second-order linear differential equations are obtained in the exact form by writing the equations in the general operator form and finding an inverse differential operator for this general operator form.

We study the problem of optimal control with impulsive component for systems described by abstract Sobolev-type differential equations with unbounded operator coefficients in Hilbert spaces. The operator coefficient of the time derivative may be noninvertible. The main assumption is a restriction imposed on the resolvent of the characteristic operator pencil in a certain right half plane. Applications to Sobolevtype partial differential equations are discussed.

We calculate a regularized trace formula for a second order differential operator with bounded operator coefficient given in a finite interval.

The main objective of this study was to generalize a differential operator. The generalized differential operator reduced to many known operators studied by various authors. New classes containing this generalized operator were studied and characterization of these classes was obtained. Further, subordination and superordination results involving this operator were studied and obtained the sandwich theorem.

A Rota–Baxter operator of weight λ is an abstraction of both the integral operator (when λ = 0 ) and the summation operator (when λ = 1 ). We similarly define a differential operator of weight λ that includes both the differential operator (when λ = 0 ) and the difference operator (when λ = 1 ). We further consider an algebraic structure with both a differential operator of weight λ and a Rota–Baxter operator of weight λ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objects in the corresponding categories. In the commutative case, the free objects are given in terms of generalized shuffles, called mixable shuffles. In the noncommutative case, the free objects are given in terms of angularly decorated rooted forests. As a byproduct, we obtain structures of a differential algebra on decorated and undecorated planar rooted forests.

In this note the anisotropic maximal operator and anisotropic Riesz potentials generated by the generalized shift operator are investigated in the anisotropic B -Morrey space Lp,λ ,γ (R n k,+) . We prove that the anisotropic B -maximal operator Mγ is bounded on the anisotropic B -Morrey space Lp,λ ,γ (R n k,+) . Also the anisotropic B -Riesz potential R α γ is bounded from the anisotropic B -Morrey spaces Lp,λ ,γ (R n k,+) to Lq,λ ,γ (R n k,+) if and only if 1/p− 1/q = α/(|a| + (a, γ ) − λ) and 1 < p < (|a| + (a, γ )−λ)/α , and its modified version Rα γ is bounded from the anisotropic B -Morrey space to the anisotropic B -BMO space. Furthermore, we obtain some imbedding relations between the space Lp,λ ,γ (R n k,+) and the anisotropic B -Stummel-Kato class Sp,θ,γ (R n k,+) . Mathematics subject classification (2000): 42B20, 42B25, 42B35.

In this paper we prove an O'Neil-type inequality for the convolution operator (B- convolution) associated with the Laplace-Bessel differential operator. By using an O'Neil- type inequality for rearrangements we obtain a pointwise rearrangement estimate of the B- convolution. As an application, we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional B-maximal operator and B-fractional integral operator with rough kernels from the spaces Lp,γ to Lq,γ and from the spaces L1,γ to the weak spaces WLq,γ .

Many mathematical models are derived in abstract, and often infinite-dimensional, spaces. Frequently, the resulting equations are operator equations which have to be solved numerically. This implies that one has to perform finitization processes yielding operator equations over finite dimensional spaces which can be solved numerically. In the present chapter we study how these approximative solutions converge to an approximation of the solution of the original problem. First, we derive methods that allow for discrete approximations of metric spaces and of operators defined over such spaces. Second, we describe the concept of collectively compact operator approximations and investigate its fundamental properties. Finally, we apply the theory developed so far to the problem of how to solve numerically Fredholm integral equations of the second kind.

In this work, a formula for the regularized trace of second order differential operator given in a finite interval which has unbounded operator coefficients is found.

The kernel K ( x , y ) of the pseudo-differential operator h μ , a is defined. It is shown that the kernel is a continuous function. It is proved that the pseudo-differential operator h μ , a is pseudolocal. A global regularity result for an elliptic partial differential operator is obtained. An existence theorem for the weak solution of a pseudo-differential equation is established.

The paper studies the Wiener-Hopf equations with kernels representable as superposition of complex-valued exponents. Such kernels arise in the kinetic gas theory, in the radiation transfer, etc. By application of a special, three-factor expansion of the initial uninvertible operator, the solution of the considered equation is reduced to those of two simple Volterra equations and a Wiener-Hopf integral equation with a contractive operator. A structural existence theorem is proved.