In this paper we use fixed point theorems to guarantee the existence of solutions for inclusions of the form Au+λu+Fu∋g, where A is a quasi-m-accretive operator defined in a Banach space, λ>0, and ...the nonlinear perturbation F satisfies some suitable conditions. We apply the obtained results, among other things, to guarantee the existence of solutions of boundary value problems of the type −Δρ(u(x))+λu(x)+Fu(x)=g(x), x∈Ω, and ρ(u)=0 on ∂Ω, where the Laplace operator Δ should be understood in the sense of distributions over Ω and to study the existence and uniqueness of solution for a nonlinear integro-differential equation posed in L1(Ω).
A high-order wavelet integral collocation method (WICM) is developed for general nonlinear boundary value problems in physics. This method is established based on Coiflet approximation of multiple ...integrals of interval bounded functions combined with an accurate and adjustable boundary extension technique. The convergence order of this approximation has been proven to be N as long as the Coiflet with N−1 vanishing moment is adopted, which can be any positive even integers. Before the conventional collocation method is applied to the general problems, the original differential equation is changed into its equivalent form by denoting derivatives of the unknown function as new functions and constructing relations between the low- and high-order derivatives. For the linear cases, error analysis has proven that the proposed WICM is order N, and condition numbers of relevant matrices are almost independent of the number of collocation points. Numerical examples of a wide range of nonlinear differential equations in physics demonstrate that accuracy of the proposed WICM is even greater than N, and most interestingly, such accuracy is independent of the order of the differential equation to be solved. Comparison to existing numerical methods further justifies the accuracy and efficiency of the proposed method.
In this paper, we propose a fractional formulation, in terms of the Caputo derivative, of the Blasius flow described by a non-linear two-point fractional boundary value problem on a semi-infinite ...interval. We develop a finite difference method on quasi-uniform grids, based on a suitable modification of the classical L1 approximation formula and show the consistency, the stability and the convergence. The numerical results confirm the theoretical ones. Comparisons with some recently proposed results are carried out to validate the accuracy of the obtained numerical results, and to show the efficiency and the reliability of the proposed numerical method.
On an open problem of S. Fučík Dai, Guowei; Liu, Fang
Applied mathematics letters,
September 2020, 2020-09-00, Letnik:
107
Journal Article
Recenzirano
Consider the boundary value problem u′′+u=αu−+p(t), u(0)=0=u(π), α≤0. We show that a necessary and sufficient condition for the problem to be solvable is that ∫0πp(t)sintdt≥0. We thus answer ...positively a counter part of a long-standing open problem posed by S. Fučík. A more general problem is also considered and the sufficient condition for the existence of solution is obtained.
ANISOTROPIC DISCRETE BOUNDARY VALUE PROBLEMS Hammouti, Omar; Taarabti, Said; Agarwal, Ravi P.
Applicable analysis and discrete mathematics,
04/2023, Letnik:
17, Številka:
1
Journal Article
Recenzirano
Odprti dostop
For an anisotropic discrete nonlinear problem with variable exponent, we demonstrate both the existence and multiplicity of nontrivial solutions in this study. The variational principle and critical ...point theory are the key techniques employed here.
In this article, a two-dimensional nonlinear boundary value problem which is strongly related to the well-known Gelfand–Bratu model is solved numerically. The numerical results are obtained by ...employing three different numerical strategies namely: finite difference based method, a Newton multigrid method and a nonlinear multigrid full approximation storage (FAS). We are able to handle the difficulty of unstable convergence behaviour by using MINRES method as a relaxation smoother in multigrid approach with an appropriate sinusoidal approximation as an initial guess. A comparison, in terms of convergence, accuracy and efficiency among the three numerical methods demonstrate an improvement for the values of λ ∈ (0, λc. Numerical results illustrate the performance of the proposed numerical methods wherein FAS-MG method is shown to be the most efficient. Further, we present the numerical bifurcation behaviour for two-dimensional Gelfand-Bratu models and find new multiplicity of solutions in the case of a quadratic and cubic approximation of the nonlinear exponential term. Numerical experiments confirm the convergence of the solutions for different values of λ and prove the effectiveness of the nonlinear FAS-MG scheme.
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