The Double Laplace–Adomian Method for Solving Certain Nonlinear Problems in Applied Mathematics
Abstract
1. Introduction
2. Technical Prerequisites
2.1. Laplace Transform: Brief Review
- Linearity property: For all real numbers and , we have
- Shifting property: For all real numbers
- Laplace transform of derivatives: Suppose that the functions are continuous for and of exponential order. Then exists, and
- (a)
- (b)
2.2. Double Laplace Transform and Basic Properties
- (i)
- The double Laplace transform of first partial derivatives is given by
- (ii)
- The double Laplace transform of n-th partial derivatives is given by
- (iii)
- The double Laplace transform of the mixed partial derivative is given by
- (iv)
- The double Laplace transform of a product of functions of one independent variable is expressed by
3. Methodology
- Decomposition of the solution: The central idea of the ADM is to decompose the NLPDE solution u into a series whose terms are generated recursively; that is,Then, every is calculated iteratively.
- Decomposition of nonlinear terms: This is a crucial part of the ADM. The nonlinear term is decomposed into an infinite series of Adomian polynomials, denoted as :These Adomian polynomials are generated in a manner that implies they are solely dependent on the components , ,…, .Kataria and Vellaisamy in [45] introduced a recursive technique for computing Adomian polynomials, eliminating the need for laborious computations of the nth derivative. With this method, each is computed as
4. Illustrative Examples of Application
4.1. Model 1: Solitons in Biomembranes
4.2. Model 2: Ion Exchange Processes
4.3. Model 3: Schrödinger Equation with Defocusing Strength Nonlinearities
4.4. Model 4: Allen–Cahn Equation
4.5. Model 5: Nonlinear Black–Scholes Equation with Variable Volatility
5. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Explicit Calculations for Model 3: Nonlinear Schrödinger Equation
Appendix B. Explicit Calculations for Model 4: Allen–Cahn Equation
Appendix C. The Existence and Uniqueness of Solutions for the Allen–Cahn Equation
- Given , , and , we search for a solution for Equation (A1).
- for a non-negative function such that , satisfying
- Solutions generally have lower energy levels and create interfaces that distinguish regions where the values remain nearly constant, approaching the minima of F.
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Cases | c | A | B | C | K | |||||||
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2 | 14 |
Case | K-Steps | DLADM | ADM | |||
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1 | 14 |
x | Absolute Error at | Absolute Error at | Absolute Error at |
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x | Absolute Error at | Absolute Error at | Absolute Error at |
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x | Absolute Error at | Absolute Error at | Absolute Error at |
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González-Gaxiola, O. The Double Laplace–Adomian Method for Solving Certain Nonlinear Problems in Applied Mathematics. AppliedMath 2025, 5, 98. https://doi.org/10.3390/appliedmath5030098
González-Gaxiola O. The Double Laplace–Adomian Method for Solving Certain Nonlinear Problems in Applied Mathematics. AppliedMath. 2025; 5(3):98. https://doi.org/10.3390/appliedmath5030098
Chicago/Turabian StyleGonzález-Gaxiola, Oswaldo. 2025. "The Double Laplace–Adomian Method for Solving Certain Nonlinear Problems in Applied Mathematics" AppliedMath 5, no. 3: 98. https://doi.org/10.3390/appliedmath5030098
APA StyleGonzález-Gaxiola, O. (2025). The Double Laplace–Adomian Method for Solving Certain Nonlinear Problems in Applied Mathematics. AppliedMath, 5(3), 98. https://doi.org/10.3390/appliedmath5030098