Nonlinear Parabolic Boundary Value Problems Research Articles

Solvability of the Quaternary Continuous Classical Boundary Optimal Control Dominated by Quaternary Parabolic System

The purpose of this paper is to study the solvability of the quaternary continuous classical boundary optimal control vector problem dominated by quaternary nonlinear parabolic boundary value problem with state constraints. The existence theorem for a quaternary continuous classical boundary optimal control vector with equality and inequality state constraints is stated and demonstrated under suitable assumptions. The mathematical formulation of the quaternary adjoint eqs. associated with the quaternary nonlinear parabolic boundary value problem with state constraints is discovered. The Fréchet derivative of the cost function and the constraints functions are derived. The necessary and sufficient theorems (conditions) for optimality of the quaternary continuous classical boundary optimal control vector problem are stated and demonstrated under suitable assumptions.

Quaternary Boundary Optimal Control Problem Dominating by Quaternary Nonlinear Parabolic System

In this paper, our purpose is to study the quaternary continuous classical boundary optimal control vector problem (QCCBOCVP) dominated by the quaternary nonlinear parabolic boundary value problem (QNLPBVP). Under suitable assumptions and with given quaternary continuous classical boundary control vector (QCCBCV), the existence theorem for a unique quaternary state vector solution (QSVS) of the weak form (WF) for the QNLPBVP is stated and demonstrated via the Method of Galerkin and the first compactness theorem. Furthermore, the continuity of the Lipchitz operator between the QSVS of the WF for the QLPBVP and the corresponding QCCBCV is proved. The existence of a quaternary continuous classical boundary optimal control vector (QCCBOVC) is stated and demonstrated under suitable assumptions.

Classical Continuous Constraint Boundary Optimal Control Vector Problem for Triple Nonlinear Parabolic System

In this paper, our purpose is to study the classical continuous constraints boundary optimal triple control vector problem dominating nonlinear triple parabolic boundary value problem. The existence theorem for a classical continuous triple optimal control vector CCCBOTCV is stated and proved under suitable assumptions. The mathematical formulation of the adjoint triple boundary value problem associated with the nonlinear triple parabolic boundary value problem is discovered. The Fréchet derivative of the Hamiltonian is derived. Under proper assumptions, both theorems are granted; the necessary conditions for optimality and the sufficient conditions for optimality of the classical continuous constraints boundary optimal triple control vector problem are stated and proven.

Classical Continuous Boundary Optimal Control Vector Problem for Triple Nonlinear Parabolic System

In this paper, our purpose is to study the classical continuous boundary optimal triple control vector problem (CCBOTCVP) dominating by nonlinear triple parabolic boundary value problem (NLTPBVP). Under suitable assumptions and with given classical continuous boundary triple control vector (CCBTCV), the existence theorem for a unique state triple vector solution (STVS) of the weak form W.F for the NLTPBVP is stated and demonstrated via the Method of Galerkin (MGa), and the first compactness theorem. Furthermore, the continuity operator between the STVS of the WFO for the NLTPBVP and the corresponding CCBTCV is stated and demonstrated. The continuity of the Lipschitz (LIP.) operator between the STVS of the WFo for the QNLPBVP and the corresponding CCBTCV is proved. The existence of a CCBOTCV is stated and demonstrated under suitable conditions.

The Classical Continuous Optimal Control for Quaternary Nonlinear Parabolic Boundary Value Problems

In this paper, our purpose is to study the classical continuous optimal control (CCOC) for quaternary nonlinear parabolic boundary value problems (QNLPBVPs). The existence and uniqueness theorem (EUTh) for the quaternary state vector solution (QSVS) of the weak form (WF) for the QNLPBVPs with a given quaternary classical continuous control vector (QCCCV) is stated and proved via the Galerkin Method (GM) and the first compactness theorem under suitable assumptions(ASSUMS). Furthermore, the continuity operator for the existence theorem of a QCCCV dominated by the QNLPBVPs is stated and proved under suitable conditions.

Existence of renormalized solutions to a nonlinear parabolic boundary value problem with a singular lower order term and a nonhomogeneous datum

Existence of renormalized solutions to a nonlinear parabolic boundary value problem with a singular lower order term and a nonhomogeneous datum

The Classical Continuous Optimal Control for Quaternary Nonlinear Parabolic Boundary Value Problems with State Vector Constraints

This paper aims to study the quaternary classical continuous optimal control problem consisting of the quaternary nonlinear parabolic boundary value problem, the cost function, and the equality and inequality constraints on the state and the control. Under appropriate hypotheses, it is demonstrated that the quaternary classical continuous optimal control ruling by the quaternary nonlinear parabolic boundary value problem has a quaternary classical continuous optimal control vector that satisfies the equality constraint and inequality state and control constraint. Moreover, mathematical formulation of the quaternary adjoint equations related to the quaternary state equations is discovered, and then the weak form of the quaternary adjoint equations is obtained. Lastly, both the necessary conditions for optimality and sufficient conditions for optimality of the proposed problem are stated and proved. The derivation for the Fréchet derivative of the Hamiltonian is attained.

The Optimal Control Problem for Triple Nonlinear Parabolic Boundary Value Problem with State Vector Constraints

In this paper, the classical continuous triple optimal control problem (CCTOCP) for the triple nonlinear parabolic boundary value problem (TNLPBVP) with state vector constraints (SVCs) is studied. The solvability theorem for the classical continuous triple optimal control vector CCTOCV with the SVCs is stated and proved. This is done under suitable conditions. The mathematical formulation of the adjoint triple boundary value problem (ATHBVP) associated with TNLPBVP is discovered. The Fréchet derivative of the Hamiltonian" is derived. Under suitable conditions, theorems of necessary and sufficient conditions for the optimality of the TNLPBVP with the SVCs are stated and proved.

An existence result for a strongly nonlinear parabolic equations with variable nonlinearity

We prove the existence of a solution for the strongly nonlinear parabolic initial boundary value problem associated to the equation ut − div a(x, t, ∇u) + g(x, t, u, ∇u) = f, where the vector field a(x, t, ξ) exhibits non-standard growth conditions.

Nonlinear systems of parabolic IBVP: A stable super-supraconvergent fully discrete piecewise linear FEM

The main purpose of this paper is the design of discretizations for second order nonlinear parabolic initial boundary value problems which are stable and present second order of convergence with respect to H1-discrete norms. The convergence results are established assuming that the solutions are in H3. The stability analysis of numerical methods around a numerical solution requires the uniform boundness of such solution. Although such bounds are usually taken as an assumption, in this paper these will be deduced as a corollary of suitable error estimates. As the methods can be simultaneously seen as piecewise linear finite element methods and finite difference methods, the convergence results can be seen simultaneously as superconvergence results and supraconvergence results. Numerical results illustrating the sharpness of the smoothness assumptions and an application to simulation of the solar magnetic field in the umbra (the central zone of a sunspot) are also included.

Similarity solutions of fractional parabolic boundary value problems with uncertainty

We use similarity method to solve nonlinear fractional order fuzzy parabolic boundary value problem with Caputo fractional derivative. The fractional order fuzzy parabolic equation is converted into fuzzy fractional differential equation. Schauder’s fixed point theorem is used to derive the existence results. Finite difference method is used to obtain the numerical results. Examples are provided to justify the results.

Cubature rules based on a bivariate degree-graded alternative orthogonal basis and their applications

The purpose of this paper is to derive cubature formulas using bivariate degree-graded alternative shifted Jacobi polynomials and present some of their applications through a method for fast and explicit construction of operational integration matrices in this polynomial basis. First, we take the non-degree-graded alternative shifted Jacobi polynomial basis over the interval [0,1] and introduce a corresponding degree-graded polynomial basis. We then construct sparse and well-structured operational integration matrices in that basis. The main advantage of the proposed idea is the low computational complexity which is derived through vector–matrix multiplications without change of bases. As an application, a fast and accurate numerical algorithm based on the obtained cubature formula is developed for the approximate solution of nonlinear multi-dimensional integral equations which arise in the theory of nonlinear parabolic boundary value problems as well as the mathematical modeling of the spatio-temporal development of an epidemic. It is shown that such a matrix representation of cubature formula in terms of the bivariate degree-graded basis gives an approximate solution with a higher order accuracy. The experimental results also illustrate that smaller size and lower orders of the operational matrix and basis functions can obtain a favorable approximate solution.

Existence of weak solutions for a nonlinear parabolic equations by Topological degree

We study the nonlinear parabolic initial boundary value problem associated to the equation ut − diva(x, t, u, grad u) = f(x, t), where the terme − diva(x, t, u, grad u) is a Leray-Lions operator, The right-hand side f is assumed to belong to L^q(Q). We prove the existence of a weak solution for this problem by using the Topological degree theory for operators of the form L + S, where L is a linear densely defined maximal monotone map and S is a bounded demicontinuous map of class (S+) with respect to the domain of L.

Numerical Verification of Solutions for Nonlinear Parabolic Problems

In this paper, we present a numerical verification method of solutions for nonlinear parabolic initial boundary value problems. Decomposing the problem into a nonlinear part and an initial value part, we apply Nakao’s projection method, which is based on the full-discrete finite element method with constructive error estimates, to the nonlinear part and use the theoretical analysis for the heat equation to the initial value part, respectively. We show some verified examples for solutions of nonlinear problems from initial value to the neighborhood of the stationary solutions, which confirm us the actual effectiveness of our method.

Superconvergent discontinuous Galerkin methods for nonlinear parabolic initial and boundary value problems

Abstract In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence $\begin{array}{} \displaystyle O\big(\!\sqrt{{}\log(T/h^2)}\,h^{k+2}\big) \end{array}$ in L∞(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems.

On some discrete qualitative properties of implicit finite difference solutions of nonlinear parabolic problems

In this paper we investigate two special qualitative properties of the finite difference solutions of one-dimensional nonlinear parabolic initial boundary value problems. The first property says that the number of the so-called L-level points, or specially the number of the zeros, of the solution must be non-increasing in time. The second property requires a similar property for the number of the local maximizers and minimizers. First we recall a theorem that guarantees the above properties for the solution of a special second order nonlinear parabolic problem. Then we generate the numerical solution with the implicit Euler finite difference method and show that the obtained numerical solution satisfies the discrete versions of the above properties without any requirements on the mesh parameters. We close the paper with some numerical tests.

The Approximation Solution of a Nonlinear Parabolic Boundary Value Problem Via Galerkin Finite Elements Method with Crank-Nicolson

This paper deals with finding the approximation solution of a nonlinear parabolic boundary value problem (NLPBVP) by using the Galekin finite element method (GFEM) in space and Crank Nicolson (CN) scheme in time, the problem then reduce to solve a Galerkin nonlinear algebraic system(GNLAS). The predictor and the corrector technique (PCT) is applied here to solve the GNLAS, by transforms it to a Galerkin linear algebraic system (GLAS). This GLAS is solved once using the Cholesky method (CHM) as it appear in the matlab package and once again using the Cholesky reduction order technique (CHROT) which we employ it here to save a massive time. The results, for CHROT are given by tables and figures and show the efficiency of this method, from other sides we conclude that the both methods are given the same results, but the CHROT is very fast than the CHM.

Existence of periodic solutions for some quasilinear parabolic problems with variable exponents

In this paper, we prove the existence of at least one periodic solution for some nonlinear parabolic boundary value problems associated with Leray–Lions’s operators with variable exponents under the hypothesis of existence of well-ordered sub- and supersolutions.

Nonlinear parabolic boundary value problems of infinite order

In this paper an existence result is presented for solution of a parabolic boundary value problem under Dirichlet null boundary conditions for a class of general equations of infinite order with strongly nonlinear perturbation terms.

Numerical solution for three-dimensional nonlinear mixed Volterra–Fredholm integral equations via three-dimensional block-pulse functions

We are concerned here with a three-dimensional nonlinear mixed Volterra–Fredholm integral equations of the second kind which include many key integral that appear in the theory of nonlinear parabolic boundary value problems. The existence of a unique solution will be proved. A new numerical method for solving these type of equations will be presented. The method is based upon three-dimensional block-pulse functions approximation. In addition convergence analysis of the method is discussed. Illustrative examples are included to demonstrate the validity and applicability of the technique.