\(n\)-dimensional differential transformation method for solving PDEs. (English) Zbl 1065.35011

Generalization of the differential transformation method [see C. K. Chen, S. H. Ho, Appl. Math. Comput. 79, 173–188 (1996; Zbl 0879.34077)] to the \(n\)-dimensional case is considered. The differential transform of the function \(w(x),\,(x\in D\subset\mathbb R^n)\) is defined as \(W(k)=\dfrac{1}{k!}\dfrac{\partial^{| k| }} {\partial x^k}w(x)\), where \(k=(k_1,\dotsc,k_n)\), and inverse transform is defined as \(w(x)=\sum_{k=0}^{\infty}W(k)x^k\). Some operations of the differential transform method are generalized too: Theorem 1 states that if \(w(x)=u(x)\cdot v(x)\), then \(W(k)=\sum_{\alpha=0}^{k} U(\alpha_1,k_2-\alpha_2,\dotsc,k_n- \alpha_n)V(k_1-\alpha_1, \alpha_2,\dotsc,\alpha_n)\). The application of the proposed method to solve some boundary and initial value problems is shown in three examples. One of these is the initial value problem to the nonlinear parabolic equation \(\alpha\partial u/\partial t +u\Delta u=0\), \(u(x,y,0)=x^2+y^2\).


35A22 Transform methods (e.g., integral transforms) applied to PDEs
35C05 Solutions to PDEs in closed form
35K55 Nonlinear parabolic equations


Zbl 0879.34077
Full Text: DOI


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