In this article, by the fixed point theorem in a cone and the nonlocal fourth-order BVP's Green function, the existence of at least one positive solution for the nonlocal fourth-order boundary value problem with all order derivatives u ( 4 ) ( t ) + A u ″ ( t ) = λ f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) , u ‴ ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = ∫ 0 1 p ( s ) u ( s ) d s , u ″ ( 0 ) = u ″ ( 1 ) = ∫ 0 1 q ( s ) u ″ ( s ) d s is considered, where f is a nonnegative continuous function, λ > 0, 0 < A < π 2, p, q ∈ L[0, 1], p(s) ≥ 0, q(s) ≥ 0. The emphasis here is that f depends on all order derivatives.

By using a fixed point theorem in a cone and the nonlocal third-order BVP's Green function, the existence of at least one positive solution for the third-order boundary-value problem with the integral boundary conditionsx′′′(t)+f(t,x(t),x′(t))=0,t∈J,x(0)=0,x′′(0)=0, andx(1)=∫01g(t)x(t)dtis considered, wherefis a nonnegative continuous function,J=[0,1], andg∈L[0,1].The emphasis here is thatfdepends on the first-order derivatives.

In this paper, by the use of a new fixed point theorem and the Green function of BVPs, the existence of at least one positive solution for the third-order boundary value problem with the integral boundary conditions is considered,where there is a nonnegative continuous function. Finally, an example which to illustrate the main conclusions of this paper is given.

In this paper, a new complex waves representing solutions of the complex Ginzburg–Landau equation with Kerr law nonlinearity is investigated. we used the extended rational sin–cos and sinh–cosh function methods construct precise solutions to the nonlinear equation. Bright periodic solution, periodic solution, dark wave soliton solution of phenomena that occur in nonlinear optics or in plasma physics are obtained. The physical meaning of the geometrical structures for some solutions is discussed for different choices of the free parameters. The proposed method provides an important and effective mathematical tool to construct exact solutions according to different complex equations. The results present the dynamics properties of the different waves with different the 3D and contour plots.

In order tofurther supplement and refine the energy theory for high-strain-rate material testing, this study systematically investigates the dynamic mechanical response and energy evolution of materials using the Split Hopkinson Pressure Bar (SHPB) technique. A computational model for key energy parameters during dynamic loading is established through theoretical derivation and subsequently validated experimentally using four representative materials. The research demonstrates that the product of the internal absorbed energy density of the material and the specimen volume equals the total absorbed energy measured by the SHPB system. This relationship can serve as an effective indicator for assessing the stress balance state of the experimental system. Experimental data show excellent agreement with theoretical predictions, with all systematic errors remaining below 3%, thereby confirming the accuracy and reliability of the theoretical derivation of energy parameters. The findings provide a significant theoretical framework and technical foundation for evaluating the dynamic mechanical properties of materials in fields such as aerospace and deep-earth exploration.