A new analytical approach for solving nonlinear boundary value problems arising in nonlinear phenomena

Abstract

In this research a new analytical approach is used to solve nonlinear boundary value problems (BVPs) of higher order occurring in nonlinear phenomena. It converts a complex nonlinear problem into zeroth order and first order problem. It consists of initial guess, auxiliary functions (containing unknown convergence controlling parameters) and a homotopy. The unknown parameters are determined by minimizing the residual. Many methods which are explained in this paper are used to determine these parameters. Here Galerkin?s method is used for this purpose. It is applied to solve non-linear BVPs of fourth and fifth order. The results are compared with the already existing methods e.g., Galerkin?s Method with Quintic B-splines, Differential Transform Method (DTM), and Optimal Homotopy AsymptoticMethod (OHAM). It gives efficient and accurate first-order approximate solution. The results achieved by this technique are in excellent concurrence with the exact solution and hence proved that this method is effective and easy to apply.

Exact solutions of (2 + 1)‐dimensional Schrödinger's hyperbolic equation using different techniques

“…By balancing the highest order derivative and highest power of nonlinear term in Equation ( 7) the value of N is determined. Substituting Equation (10) into Equation ( 7) along with Equation (11) provides a system of algebraic equations for F j (j = (0,1,2,3, … , N)). By solving this system, we get set of solutions as follows.…”

“…Substituting Equation (10) into Equation ( 7) along with Equation ( 20) with value of N, provides a set of algebraic equations with the values of…”

“…Form 1: In this section, we apply the new EHFM for the solutions of (2 + 1)-dimensional SHE. Utilizing balancing principal in (3), gives N = 1, so (10) reduces to…”

“…where Φ(x, y, t) = ax + by + dt + 𝜃 0 , 𝜉 = x + 𝜇y − 𝜆t. Form 2: Utilizing balancing principal in (3), yields N = 1, so (10) reduces to where F 0 and F 1 are constants to be described. Replacing (10) into (3) and comparing the coefficients of each polynomial of Q(𝜉) to zero, we obtain a set of algebraic equations in F 0 , F 1 , G, and H. Solving the set of equations, we obtain…”

“…However, many mathematical experts have investigated different techniques by which one can reveal the exact solutions of such NPDEs. Recently, some new effective techniques have been developed like Riccati-Bernoulli sub-ODE method [6], exp-function method [7,8], optimal homotopy asymptotic method [9][10][11][12], tanh-sech method [13,14],…”