Abstract

This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. We integrate a linearization approach for nonlinear equations with the variation of parameters method and Adomian polynomials to obtain exact solutions for higher-order nonlinear boundary and initial value problems. The suggested approach comprises three part. In the initial segment of this approach, a linearization strategy is suggested to transform higher-order nonlinear ordinary differential equations into an infinite system of linear nonhomogeneous ordinary differentials utilizing the Adomian polynomial. In the second phase of this approach, these equations are resolved using the variation of parameters method. In the final segment of this approach, we merge the outcomes from earlier steps to derive a generalized iterative method for addressing the given nonlinear equation. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed form solution. The proposed approach is very efficient and essentially perfect for determining exact solutions of nonlinear equations. To demonstrate the effectiveness of this method, we examined the extended (2 + 1) dimensional equation for thermophoretic motion, which is based on wrinkle wave movements in graphene sheets supported by a substrate. The implementation of the suggested approach effectively yielded closed-form solutions in terms of exponential functions, hyperbolic functions, trigonometric functions, algebraic functions, and Jacobi elliptic functions, respectively. Three generated solutions illustrated to examine the characteristics of thermophoretic waves in graphene sheets. The proposed method’s benefits and drawbacks are also examined. Consequently, unlike previous solutions obtained via the variation of parameters method for nonlinear issues, the solutions presented here are exact and unique.

Abstract

This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. We integrate a linearization approach for nonlinear equations with the variation of parameters method and Adomian polynomials to obtain exact solutions for higher-order nonlinear boundary and initial value problems. The suggested approach comprises three part. In the initial segment of this approach, a linearization strategy is suggested to transform higher-order nonlinear ordinary differential equations into an infinite system of linear nonhomogeneous ordinary differentials utilizing the Adomian polynomial. In the second phase of this approach, these equations are resolved using the variation of parameters method. In the final segment of this approach, we merge the outcomes from earlier steps to derive a generalized iterative method for addressing the given nonlinear equation. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed form solution. The proposed approach is very efficient and essentially perfect for determining exact solutions of nonlinear equations. To demonstrate the effectiveness of this method, we examined the extended (2 + 1) dimensional equation for thermophoretic motion, which is based on wrinkle wave movements in graphene sheets supported by a substrate. The implementation of the suggested approach effectively yielded closed-form solutions in terms of exponential functions, hyperbolic functions, trigonometric functions, algebraic functions, and Jacobi elliptic functions, respectively. Three generated solutions illustrated to examine the characteristics of thermophoretic waves in graphene sheets. The proposed method’s benefits and drawbacks are also examined. Consequently, unlike previous solutions obtained via the variation of parameters method for nonlinear issues, the solutions presented here are exact and unique.

Abstract

This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. We integrate a linearization approach for nonlinear equations with the variation of parameters method and Adomian polynomials to obtain exact solutions for higher-order nonlinear boundary and initial value problems. The suggested approach comprises three part. In the initial segment of this approach, a linearization strategy is suggested to transform higher-order nonlinear ordinary differential equations into an infinite system of linear nonhomogeneous ordinary differentials utilizing the Adomian polynomial. In the second phase of this approach, these equations are resolved using the variation of parameters method. In the final segment of this approach, we merge the outcomes from earlier steps to derive a generalized iterative method for addressing the given nonlinear equation. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed form solution. The proposed approach is very efficient and essentially perfect for determining exact solutions of nonlinear equations. To demonstrate the effectiveness of this method, we examined the extended (2 + 1) dimensional equation for thermophoretic motion, which is based on wrinkle wave movements in graphene sheets supported by a substrate. The implementation of the suggested approach effectively yielded closed-form solutions in terms of exponential functions, hyperbolic functions, trigonometric functions, algebraic functions, and Jacobi elliptic functions, respectively. Three generated solutions illustrated to examine the characteristics of thermophoretic waves in graphene sheets. The proposed method’s benefits and drawbacks are also examined. Consequently, unlike previous solutions obtained via the variation of parameters method for nonlinear issues, the solutions presented here are exact and unique.

Abstract

This article proposes a novel approach for determining exact solutions to nonlinear ordinary differential equations. We integrate a linearization approach for nonlinear equations with the variation of parameters method and Adomian polynomials to obtain exact solutions for higher-order nonlinear boundary and initial value problems. The suggested approach comprises three part. In the initial segment of this approach, a linearization strategy is suggested to transform higher-order nonlinear ordinary differential equations into an infinite system of linear nonhomogeneous ordinary differentials utilizing the Adomian polynomial. In the second phase of this approach, these equations are resolved using the variation of parameters method. In the final segment of this approach, we merge the outcomes from earlier steps to derive a generalized iterative method for addressing the given nonlinear equation. The recommended iterative method provides the solution via a rapidly converging series that readily approaches a closed form solution. The proposed approach is very efficient and essentially perfect for determining exact solutions of nonlinear equations. To demonstrate the effectiveness of this method, we examined the extended (2 + 1) dimensional equation for thermophoretic motion, which is based on wrinkle wave movements in graphene sheets supported by a substrate. The implementation of the suggested approach effectively yielded closed-form solutions in terms of exponential functions, hyperbolic functions, trigonometric functions, algebraic functions, and Jacobi elliptic functions, respectively. Three generated solutions illustrated to examine the characteristics of thermophoretic waves in graphene sheets. The proposed method’s benefits and drawbacks are also examined. Consequently, unlike previous solutions obtained via the variation of parameters method for nonlinear issues, the solutions presented here are exact and unique.