An existence and uniqueness theorem for wave equations with scalar nonlinearities

We treat wave equations with “scalar nonlinearities” and demonstrate the connection between the bifurcation theory of an associated system of Hammerstein integral equations and the existence of periodic solutions of the nonlinear wave equation.

Solution of the fourth and fifth boundary value problems for a one-dimensional second-order parabolic equation in a curvilinear region

Background : The teaching of differential equations is dominated by an excessively algebraised analytic tradition. For this reason, studies that contribute to conceptualising mathematical objects associated with the differential equation are important, particularly the existence and uniqueness theorem. Objectives : From its genesis, the objective is to analyse the nature of this knowledge, its epistemology from practice. We give an account of the variational arguments , i.e. , based on practices focused on the study of change, with a predictive purpose, which allows obtaining the desired result on the differential equation: demonstrating the existence of a unique solution. Design : A documentary analysis is carried out from the Socioepistemological Theory of the works that marked the construction of this mathematical knowledge. Setting and Participants : Being a documentary-cut study, we did not have participants stricto sensu . Data collection and analysis : Our observation unit includes mathematical works as primary and secondary sources involved in constructing the theorem: its postulations, search for hypotheses and proofs. Results : A reconstruction of the theorem is offered, which from the arguments, characterises some practices that helped in the construction of mathematical objects. Conclusions : We conclude that the bounded variation, as a particular way of using change, contributed to the search or establishment of conditions for the interpretation of the solution of equations and to obtain a unique solution to the differential equation, contributions that should be key for implementations of learning situations.

Uncertain delay differential equations are a type of differential equations driven by a Liu process. So far, it has been proved that uncertain delay differential equation has a unique solution in the finite domain, in which its coefficients satisfy the global Lipschitz continuity. This paper continues to focus on the existence and uniqueness of the solutions of uncertain delay differential equations on the infinite domain. Meanwhile, a new existence and uniqueness theorem for uncertain delay differential equations under one-sided local Lipschitz condition and linear growth condition is deduced.

The focus of this paper is on student interpretation and usage of the existence and uniqueness theorems for first-order ordinary differential equations. The inherent structure of the theorems is made explicit by the introduction of a framework of layers concepts–conditions–connectives–conclusions, and we discuss the manners in which students’ understanding of these layers construct their perception of the theorems as a whole.

The problem of determining source term in a semilinear wave equation is considered. The source term is represented as the product ƒ(u(x, t))p(x), where ƒ(s) is a given function, u(x, t) is a solution to Cauchy problem for wave equation, p(x) is an unknown function. To determine p(x) the additional information on the solution of the Cauchy problem u(α(t), t) = g(t), u(β(t), t) = h(t) is used. Theorems of existence and uniqueness of solution to an inverse problem in the class of continuous functions p(x) and in the class of functions p(x) = po + xq(x), where q(x) is continuous, are proved.

The problem of determining source term in a semilinear wave equation is considered. The source term is represented as the product ƒ(u(x, t))p(x), where ƒ(s) is a given function, u(x, t) is a solution to Cauchy problem for wave equation, p(x) is an unknown function. To determine p(x) the additional information on the solution of the Cauchy problem u(α(t), t) = g(t), u(β(t), t) = h(t) is used. Theorems of existence and uniqueness of solution to an inverse problem in the class of continuous functions p(x) and in the class of functions p(x) = po + xq(x), where q(x) is continuous, are proved.

An existence theorem is proved for closed convex surfaces whose principal radii of curvature regarded as functions of the unit normal vectorn satisfy the equation $$R_1 R_2 + \Phi (R_1 + R_2 ,R_1 ,R_2 ,n) + c \cdot n = \varphi (n),$$ wheree is an unknown constant vector being sought along with the surface. This equation has been studied by Miranda, who proved a variety of existence and uniqueness theorems about its solutions. The results obtained lead to a certain simplification in the hypotheses imposed on the functions Φ andϕ in these theorems.

One normal problem of the field theory on a plane for a system of Poisson equations is studied. This problem is one of implementations of the general approach to formulation of boundary value problems of the field theory, which is based on studying the kernels of trace operators and kernels of trace functionals, which are taken as the main space of desired solutions. A peculiarity of this problem is that one of the boundary conditions imposed on the desired solution is nonlocal in nature. The considered boundary condition nonlocality is due to the fact that the base space in which the solution of the problem is sought is the kernel of the regular trace functional in the Sobolev space. The need to supplement the problem on the one-dimensional cokernel of the above-mentioned functional, which arises in this case, determines the second boundary condition, which is local (pointwise) in nature. That is why, in the framework of the duality of the Sobolev space and its conjugate space, the solution to the problem is the pair (u, с), where с is a constant that compensates for the one-dimensional “defect” of the problem. It should be noted that the theory of boundary value problems for vector fields on a plane differs essentially from the theory of boundary value problems for three-dimensional vector fields. One of the reasons for the difficulties in formulating three-dimensional boundary value problems is that a number of important surfaces cannot be made parallel to each other, which rules out the possibility to consider “tangential” boundary value problems, that is, problems with boundary tangential fields. In this respect, the theory of boundary value problems of plane vector fields differs from the three-dimensional theory for the better. On the other hand, the fact that a formula that would interrelating the Laplace operator with first-order field theory operations is not available on a plane is an obstacle to identifying new non-standard boundary value problems for plane fields. Revealing of this interrelation is a separate issue, the solution of which leads to the concept of the trace of first-order field theory operators as a singular functional above the space of traces of functions from the first-order Sobolev space. Thus, boundary value problems for plane fields are a separate section of the general theory of field boundary value problems. One of the non-standard problems in the theory of plane vector fields, which contains non-local boundary conditions, is considered. A theorem for existence and uniqueness of a weak solution of the problem posed is formulated and proved. The proof of the theorem includes a few steps: using the Galerkin method, the existence of a solution to the equation is established; the passage to the limit is made, and the vector function u(x) is found; the number α is determined, and the uniqueness of the function u and the number α is established. The main result of the work is that the problem posed is correct in nature.

(1964). Existence and Uniqueness Theorems for Solutions of Difference Equations. The American Mathematical Monthly: Vol. 71, No. 8, pp. 859-862.

Recently, a new fractional derivative called the conformable fractional derivative is given on based basic limit definition derivative in [4]. Then, the fractional versions of chain rules, exponential functions, Gronwalls inequality, integration by parts, Taylor power series expansions is developed in [5]. In this paper, we present existence and uniqueness theorems for sequential linear conformable fractional differential equations.

The inverse problems of finding the free term and the coefficient of u(x,t) in a parabolic equation is considered. The Fredholm property of the linear inverse problem of finding the right-hand side of a special form is proved, along with global existence, uniqueness, and stability theorems for its solutions. A uniqueness theorem is established for the nonlinear inverse problem of determining the coefficient under restrictions in the form of inequalities containing no smallness conditions. The proof is carried out by a method of a priori estimates, with the use of the maximum principle for parabolic and elliptic equations. A connection between the uniqueness of the solution of the inverse problem and the completeness of a certain system of functions is established.

Existence and uniqueness theorems for periodic Markov process and applications to stochastic functional differential equations

The inverse problem for a semilinear functional-differential wave equation is studied. The theorems of existence and uniqueness of solution to the inverse problem are proved.

Free boundary problems for the heat equation in which the moving interface coincides initially with the fixed face