Non-Linear New Product $A^{\ast }B-B^{\ast }A$ Derivations on $\ast$-Algebras

Let $\mathcal{A}$ be a prime $\ast$-algebra. In this paper, we suppose that $\mathrm{\Phi }:\mathcal{A}\to \mathcal{A}$ satisfies $$\mathrm{\Phi }(A\diamond B)=\mathrm{\Phi }(A)\diamond B+A\diamond \mathrm{\Phi }(B)$$ where $A\diamond B=A^{\ast }B-B^{\ast }A$ for all $A,B\in \mathcal{A}$ .We will show that if $\mathrm{\Phi }(\alpha \frac{I}{2})$ is self-adjoint for $\alpha \in \{1,i\}$ then $\mathrm{\Phi }$ is additive $\ast$-derivation.