N-dimensional Laplace transformations and their applications in partial differential equations (original) (raw)

CHAPTER 3. FURTHER NEW RESULTS ON N-DEMENSIONAL LAPLACE AND INVERSE LAPLACE TRANSFORMATIONS 3.1. Introduction 3.2. The Image of Functions with the Argument ' 89 3.2.1. Applications of Theorem 3.2.1 3.2.2. Laplace Transforms of some Elementary and Special Functions with n Variables ^ 98 3.3. The Original of Functions with the Argument ^ 106 3.3.1. Examples Based Upon Theorem 3.3.1 Ill 3.4. The Image of Functions with the Argument 2pi(x 114 3.4.1 Applications of Theorem 3.4.1 119 3.5. The Original of Functions with the Argument 121 3.5.1. Example Based Upon Theorem 3.5.1 125 CHAPTER 4. THE SOLUTION OF INITIAL-BOUNDARY-VALUE PROBLEMS (IBVP'S) BY DOUBLE LAPLACE TRANSFORMATIONS 127 4.1. Introduction 127 4.2. Non-homogenous Linear Partial Differential Equations (PDEs) of the First Order 129 4.2.1. Partial Differential Equations of Type Ux+u.y = f{x,y),Q<x<«o,0<y<oo 129 4.2.2. Partial Differential Equations of Type au^ + buy + eeu = fix,y), 0<a:<<», 0<y<~ 136 4.3. Non-homogenous Second Order Linear Partial Differential V Equations of Hyperbolic Type 4.3.1. Partial Differential Equations of Type Ujcy = f(x,y),),0<x«>o,0<y<oo 4.3.2. The Wave Equation 4.4. Non-homogenous Second Order Partial Differential Equations of Parabolic Type 4.4.1. Partial Differential Equations of Type u"+2u^+Uyy + KU = fix,y\0<x<oo,0<y<oo CHAPTER 5. CONCLUSIONS AND FUTURE DIRECTIONS 154 5.1. Conclusions 154 5.2. Future Directions 155