In the present paper, we precisely conduct a quantum calculus method for the numerical solutions of PDEs. A Nonlinear Schrödinger equation is considered. Instead of the known classical discretization methods based on the finite difference scheme, Adomian method, and third modified ver-sions, we consider a discretization scheme leading to subdomains according to q-calculus and provide an approximate solution due to a specific value of the parameter q. Error estimates show that q-calculus may produce effi-cient numerical solutions for PDEs. The q-discretization leads effectively to higher orders of convergence provided with faster algorithms. The numer-ical tests are applied to both propagation and interaction of soliton-type solutions.

This paper is concerned with a version of Saint-Venant type decay estimates for solutions of the Dirichlet problem for the Nonlinear biharmonic equation. Energy methods are used to establish a Nonlinear integro-differential inequality for a quadratic functional. To this end, integral methods are used to establish exponential decay of solutions for the Dirichlet problem being considered.

This paper studies advanced mathematical methods like sin-cos and sinh-cosh approachesto find precise solutions by considering inter-modal dispersion and spatio-temporal dynamics with kerrlaw Nonlinearity in the Resonant shrödinger equation. These methods are useful for solving Nonlinearpartial differential equations. To obtain the ordinary differential equation for the traveling wave solution,we initially deal with the general partial differential equation (PDE). Then, a series of optical solitonsolutions, including cusp and dark solitons, are derived for the Resonant shrödinger equation using thiseffective approach.

The current manuscript is concerned with extracting an analytical approximate periodic solution of a damped cubic Nonlinear Klein-Gordon equation. The Riemann-Liouville fractional calculus is utilized to obtain an analytic approximate solution. The Homotopy technique is absorbed in the multiple time-spatial scales. The approved scheme yields a generalization of the Homotopy equation; whereas, two different small parameters are adapted. The first parameter concerns with the temporal perturbation, simultaneously, the second one is accompanied by the spatial one. Therefore, the analytic approximate solution needs the two perturbation expansions. This approach conducts more advantages in handling the classical multiple scales method. Furthermore, the initial conditions are included throughout the multiple scale method to achieve a special solution of the governing equation of motion. The analysis ends up deriving two first-order equations within the extended variables and their actual solution is achieved. The procedure adopted here is very promising and powerful in managing similar numerous Nonlinear problems arising in physics and engineering. Furthermore, the linearized stability of the corresponding ordinary Duffing differential equation is analyzed. Additionally, some phase portraits are shown.

The general Nonlinear fourth order differential equation, x"""" + f(1) (x,x"x "",x""")x""" + f(2) (x,x",x "")x "" + f(3)(x,x")x" + f(4)(x,x\",x "",x""",t)x = p(t) is considered. It is proved that if the functions f(i) accept certain sufficient conditions, then there exists a bounded space for state variables, in which the equation has periodic answer. For this purpose we use the existence of Green"s function and Schauder"s theorem.